LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 


MRS/  MARTHA   E.   HALLIDIE. 
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WORKS  OF 
PROFESSOR  C.  W.  MACCORD 

PUBLISHED    BY 

JOHN   WILEY   &  SONS. 


Descriptive  Geometry. 

With  Applications  to  Isometrical  Drawing  and  Cavalier 
Projection.  8vo.  vi  -f-  248  pages,  280  figures.  Cloth, 
fS.OO. 

Kinematics;   or,  Practical  Mechanism. 

8vo.      xi-{-335  pages,  306  figures.      Cloth,  $5.00. 

Mechanical  Drawing. 

PART    I.     Progressive  Exercises. 

PART  II.     Practical  Hints  for  Draughtsmen. 

The  two  parts  complete  in  one  volume.  410.  258 
pages,  232  figures.  Cloth,  $4.00. 

Velocity  Diagrams. 

Their  Construction  and  their  Uses.  Addressed  to  all 
those  interested  in  Mechanical  Movements.  Svo.  iii  + 
ub  pp.,  83  figures.  Cloth,  $1.50. 


VELOCITY    DIAGRAMS. 

THEIR  CONSTRUCTION  AND  THEIR  USES. 

INTENDED  FOR  ALL    WHO   ARE  INTERESTED   IN 
MECHANICAL   MOVEMENTS. 


BY 

CHARLES  WILLIAM  MAcCORD,  A.M.,  Sc.D., 

Professor  of  Mechanical  Drawing  in  the  Stevens  Institute  of  Technology^ 

Hoboken,  N. /.; 

Formerly  Chief  Draughtsman  for  Capt.  John  Ericsson  / 

Author  of  " Kinematics^  "  Mechanical  Drawing" 

''Elements  of  Descriptive  Geometry, "  and 

Numerous  Monographs  on  Mechanism. 


FIRST   EDITION. 
FIRST    THOUSAND. 


NEW  YORK: 

JOHN   WILEY   &   SONS. 

LONDON:    CHAPMAN  &  HALL,  LIMITED. 

1901. 


Copyright,  1901, 

BY 
CHARLES   WILLIAM    MAcCORD. 


ROBERT    DRUMMOND,    PRINTER,    NEW   YORK. 


PREFACE. 


THIS  treatise  is  in  effect  an  abstract  of  a  series  of  lectures 
forming  a  part  of  the  course  of  instruction  at  the  Stevens 
Institute  of  Technology.  It  explains  the  principles  of  the 
more  common  and  convenient  graphic  processes  of  deter- 
mining at  any  given  instant  the  direction  and  velocity  of  the 
motion  of  a  point,  whether  that  motion  be  constant  or  vari- 
able. 

It  is  intended  not  only  for  use  in  the  classroom,  but  for 
the  benefit  of  those  who  may  wish  to  study  the  subject 
without  the  aid  of  an  instructor. 

C.  W.   MACCORD. 

HOBOKEN,    N.   J.,   Oct.    12,    IQOI. 


VELOCITY   DIAGRAMS. 


1.  It  is  a  familiar  fact  that  in  the  operation  of  any  piece 
of  mechanism,  the  parts  go  through  a  series  of  motions  in 
regular  order,  finally  returning  to  their  original  positions  ; 
after  which  the  same  series  of  motions  is  repeated,  and  so 
on  indefinitely.  One  complete  series  is  called  a  cycle;  in 
completing  which  it  frequently  happens  that,  supposing  the 
first  or  driving  piece  to  move  uniformly  in  one  direction,  the 
motions  of  other  parts  will  vary  either  in  velocity  or  direc- 
tion, and  often  in  both.  And  in  studying  the  action  of  any 
mechanical  movement,  or  in  comparing  the  actions  of  differ- 
ent ones,  it  is  often  desirable  to  have  a  clear  understanding 
of  the  law  of  variation,  in  regard  to  the  motion  of  a  given 
piece  or  a  given  point. 

Now,  assuming  that  for  a  given  motion  of  the  driver,  the 
motion  of  the  point  considered  can  be  determined  at  any 
instant,  or  in  other  words  in  any  phase  of  the  action — then 
it  is  beyond  question  that  a  graphic  representation  is  the 
best  if  not  indeed  the  only  means  of  conveying  to  the  mind 
a  distinct  and  comprehensive  idea  of  the  law  according  to 
which  the  motion  varies  in  velocity  and  direction. 

Such  a  representation,  or  "  velocity  diagram/'  is  shown  in 
Fig.  I.  It  consists  merely  of  a  curve  whose  abscissas,  set 


2  VELOCITY  DIAGRAMS. 

off  from  left  to  right  upon  the  line  MN,  represent  times,  and 
the  ordinates  11',  22',  etc.,  represent  the  velocities  of  the 
moving  point  at  the  instants  indicated  by  the  points  i,  2, 
etc.;  the  positive  ordinates,  or  those  above  the  line,  indicate 
motion  in  one  direction,  that  in  the  opposite  direction  being 
indicated  by  the  negative  ordinates,  below  the  line.  And  a 
single  glance  at  this  figure  is  sufficient  to  establish  the  claim 
above  made, — it  gives  in  an  instant  all  the  information  that 
could  be  gathered  from  lengthy  explanations  and  tables  of 
figures. 


FIG.  i. 

2.  Now,  given  the  velocity  of  the  driver,  how  to  deter- 
mine the  values  of  these  ordinates  ?  We  have  here  to 
choose  between  two  methods  of  procedure,  the  graphic  and 
the  analytic.  There  is  no  question  that  the  members  of  a 
train  of  mechanism  can  be  represented  by  symbols,  the  laws 
of  their  motions  embodied  in  formulas,  and  the  desired 
values  ascertained  by  algebraic  computation.  In  the  graphic 
method,  the  motion  of  a  point  at  any  instant  is  represented 
in  magnitude  and  direction  by  a  right  line  of  definite  length  ; 
and  relations  may  be  established  between  lines  thus  repre- 
senting the  motions  of  properly  selected  points,  and  other 


VELOCITY  DIAGRAMS.  3 

lines  closely  connected  with  the  moving  pieces,  such  that 
the  values  sought  can  be  determined  by  geometric  reason- 
ing. Of  the  two,  the  latter  method  is  preferable  for  ordinary 
use,  being  far  more  simple  and  expeditious  than  the  former, 
while  the  accuracy  attainable  is  quite  sufficient  for  practical 
purposes.  Its  foundations  lie  upon  a  few  kinematic  princi- 
ples, of  which  we  will  briefly  state  the  most  important. 

3.  The  first  is  the  composition  of  motion.  Suppose  the 
point  A,  Fig.  2,  to  receive  at  the  same  instant  two  impulses, 
which  separately  would  impart  to  it  the  motions  represented 
in  direction  and  velocity  by  the  lines  AB,AC\  these  are 
called  components,,  and  the  resultant  of  these,  which  is  the 
actual  direction  and  velocity  of  the  motion  of  the  point,  is 

/y 


— x 


FIG  2. 

AD,  the  diagonal  of  the  parallelogram  ABCD.  The  condition 
of  things  here  is  that  A  is  a  free  point  in  space, — as  if,  by 
way  of  illustration,  it  were  a  billiard-ball  impelled  by  the 
simultaneous  strokes  of  two  cues  b  and  c,  which  are  not  con- 
nected with  each  other  or  with  the  ball ;  a  consideration 
which,  as  will  subsequently  appear,  is  of  no  small  importance. 
There  may  be  more  than  two  components  ;  in  that  case,  find 
the  resultant  of  any  two  of  them,  compound  that  with  any 
one  of  the  others,  and  so  on  to  the  end.  If  there  be  three 


4  VELOCITY  DIAGRAMS. 

components  not  in  the  same  plane,  these  will  be  three  ad- 
jacent edges  of  a  parallelopipedon,  and  the  resultant  will  be 
the  body  diagonal  which  passes  through  the  moving  point : 
our  attention,  however,  will  be  chiefly  confined  to  motions 
in  one  plane  or,  what  is  practically  the  same  thing,  in 
parallel  planes. 

4.  The  resolution  of  motion  is  the  exact  converse  of  the 
preceding.    If  a  motion  can  be  determined  by  compounding 
two  others,  that  motion,  if  given,  can  be  separated  or  resolved 
into  its  original  components.     Thus,  in  Fig.  2,  suppose  the 
motion  AD  to  be  assigned,  and  let  it  be  required  to  deter- 
mine  two   components   having    the  directions   Ax,  Ay,  of 
which  AD  shall  be  the  resultant.     By  drawing  through  D 
lines  parallel  to  Ax  and  Ay,  it  is  evident  that  we  shall  limit 
the  required  components,  AC,  AB.     But  AD  may  be  the 
diagonal  of  any  one  of  an  infinite  number  of  parallelograms; 
whence  it  follows  that  a  given  motion  may  be  resolved  into 
two  components  respectively  parallel  to  any  two  lines  hav- 
ing different  directions. 

5.  Side  Component  and  Longitudinal  Component. — In   Fig.  3, 
let  the  point  A,  of  the  right  line  MA,  have  a  motion  repre- 
sented by  AD.     Resolve  this  into  the  components  AC  per- 
pendicular to  MA,  and  AB  in  the  direction  of  that  line  ;  then 
AC  is  called  the  side  component,  and  AB  the  longitudinal 
component.    And  these  components  always  exist,  no  matter 
how  the  motion  AD  may  be  resolved.    Thus,  if  it  be  resolved 
into  AC',AB',  then  AC'  itself  has  a  component  Ab  along  MA, 
and  Ab  is  equal  to  B'B,  so  that  the  total  longitudinal  com- 
pgnent  is  AB'  +  Ab,  =  AB. 

Again,  in  Fig.  4,  let  AD  be  resolved  into  AE,  AF; 
then  AE  can  be  resolved  into  the  rectangular  components 
Ae,  Ag,  and  AF  into  the  pair  Af,  Ah  ;  also  Af=  eB,  and 


VELOCITY  DIAGRAMS. 


Ah  =  Cg\  Ae  and  A/lie  in  the  same  direction,  and  the  total 
longitudinal  component  is  Af-\-Ae,=AJB;  but  since  Ag 
and  Ah  lie  in  opposite  directions,  the  actual  side  component 


TIG.  4. 

is  Ag  —  Ah,  =  AC  as  before.    These  rectangular  components 
AB,  AC,  may  properly  be  designated  absolute  components. 

6.  In  Fig.  5,  let  AM  represent  a  rigid  and  inextensible 
line — as  a  piece  of  stiff  steel  wire,  and  let  the  point  A  have 
a  motion  AD,  of  which  AF  and  AE  are  the  side  and  longi- 
tudinal components  respectively.  Then  any  other  point,  B, 
of  this  wire  must  of  necessity  have  a  longitudinal  compo- 
nent BG,  equal  to  AE,  and  in  the  same  direction.  And 
whatever  the  actual  motion  of  the  point,  it  follows  from  the 
preceding  that  the  other  component  must  be  perpendicular 
to  AM,  so  that  the  extremity  of  the  resultant  must  lie  in  the 


VELOCITY  DIAGRAMS. 


indefinite  vertical  line  xx  drawn  through  G.  If,  then,  the 
direction  By  is  assigned,  the  intersection  H  of  xx  and  By 
will  determine  BH. 


0 


K 


FIG.  5. 


7.  In  Fig.  6,  AM,  AN,  represent  two  rigid  bars,  pivoted 
together  at  A  ;  let  AE,  AF,  be  the  absolute  longitudinal 
components,  from  which  it  is  required  to  find  the  -motion  of 


FIG.  6. 

A.  From  what  has  just  been  shown,  the  extremity  of  the 
resultant  must  lie  in  Ex  perpendicular  to  A M,  and  also  in 
Fy  perpendicular  to  AN\  it  must,  then,  be  their  point  of  in- 
tersection  D.  Which  is  obviously  as  it  should  be,  since  the 


VELOCITY  DIAGRAMS.  7 

resultant  AD  thus  determined  can  be  resolved  either  into 
the  rectangular  components  AE,  AG,  or  into  the  pair  AF, 
AH.  Had  we  proceeded  as  in  Fig.  i,  by  completing  the 
parallelogram  of  which  AE,  AF  are  the  sides  (as  one  not 
familiar  with  the  previous  reasoning  would  be  very  likely  to 
do),  the  diagonal  AK  would  have  neither  the  right  magni- 
tude nor  the  right  direction,  unless  AM  and  A N  were  per- 
pendicular to  each  other  :  and,  as  will  be  seen  subsequently, 
there  are  other  cases  than  this  in  which  that  procedure  will 
give  erroneous  results. 

8.  Effect  of  Side  Components. — In  Fig.  7,  let  MNbe  an  in- 
flexible steel  rod,  upon  which  the  perforated  balls  A,  B,  C, 
are  free  to  slide ;  then  the  centres  of  these  balls  are  points 
upon  the  right  line  MN,  but  not  of  it,  and,  whatever  their 
motions  may  be,  the  longitudinal  components  AH,  BL,  CR 
do  not  affect  the  line,  nor  do  they  have  any  relation  to  each 
other,  and  they  need  not  have  either  the  same  magnitude  or 
the§same  direction.  But  obviously  the  side  components  do 
affect  the  line, — if  any  two  of  them  are  equal,  and  lie  on  the 
same  side  of  MN,  the  whole  line  must  be  translated  bodily 
in  the  direction  of  those  two,  and  the  side  components  of  all 
its  points  must  be  equal.  But  if  motions  be  assigned  to  any 
two  of  these  balls,  such  that  their  side  components  either 
lie  upon  opposite  sides  of  MN  QY  are  unequal  if  they  lie 
on  the  same  side,  then  these  two  side  components  will  estab- 
lish a  rotation  of  the  line  about  some  point  of  the  line  itself. 
If,  for  example,  we  give  to  A  and  B  the  motions  AD  and  BE, 
their  side  components  are  AG  and  BK  respectively,  and, 
drawing  GK,  it  is  perceived  that  MN  must  turn  about  the 
point  O  where  GK  cuts  it.  Moreover,  this  latter  line  deter- 
mines the  directions  and  the  values  of  the  side  components 
of  the  motions  of  all  other  points  on  MN;  thus  any  point  C, 


8 


VELOCITY  DIAGRAMS. 


to  the  left  of  O,  must  move  upward,  and  that  at  a  rate  CP 
limited  by  the  prolongation  of  GK\  any  point  to  the  right  of 
O  must  move  downward  in  like  manner,  while  if  a  ball  were 
placed  at  <9,  it  could  not  move  sidewise  at  all,  though  it 


FIG.  7. 


E 


M      R 


G 


might    move    endwise    in   either   direction   and    with   any 
velocity. 

9.  Now  if,  as  in  Fig.  8,  we  consider  A,  B,  C,  to  be  points 
<?/the  inflexible  and  inextensible  line  MNt  and  therefore  as 
remaining  always  at  fixed  distances  from  each  other ;  it  is 
still  true  that  the  side  components  AG,  BK,  of  the  simul- 
taneous motions  of  two  points,  A  and  B,  for  example,  deter- 
mine the  rotation  of  MN about  O,  But  supposing  AD  to  be 
assigned,  the  motion  of  B,  for  instance,  would  no  longer  be 
entirely  arbitrary,  since  the  longitudinal  component  BL 


VELOCITY  DIAGRAMS.  9 

must  be  equal  to  AH  and  in  the  same  direction.  This  is  also 
true  of  every  point  of  MN,  including  O,  whose  absolute  mo- 
tion must  therefore  be  OQ,  =  AH.  Consequently  in  this 
case,  since  GD,  OQ,  KE,  PF,  are  all  equal,  and  all  parallel 
to  MN,  the  line  DQEF,  joining  the  extremities  of  the  result- 
ants, will  be  equal  and  parallel  to  GOKP,  joining  the  ex- 
tremities of  the  side  components. 

10.  Representation  of  Angular  Velocity. — The  linear  velocity 
of  a  point  at  unit  distance  from  a  centre  about  which  the 
point  travels  in  a  circular  arc,  is  the  measure  of  the  angular 
velocity  of  the  point  about  that  centre.  From  which  it 
follows,  that  whatever  the  distance  of  the  point  from  the 
centre,  we  shall  always  have  the  value,  angular  velocity 

linear  velocity 

radius 
If  then,  as  in  Fig.  9  the  point  A,  revolving  about  the 


centre  C,  has  the  linear  velocity  AK,  the  angle  A  CK  repre- 
sents the  angular  velocity  ;  if  it  be  desired  to  find  the  linear 
velocity  of  B  in  rotating  with  the  same  angular  velocity 
about  D,  we  have  only  to  make  the  angle  BDL  equal  to 
ACK.  If  the  angular  velocity  of  B  is  to  be  made  twice  that 
of  Ay  however,  it  is  to  be  noted  that  this  is  not  done  by 


10 


VELOCITY  DIAGRAMS. 


doubling  this  angle  ;    but  we   make  AM  =  2AK,  and  then 
make  the  angle  BDN '  =  angle  ACM. 

11.  The  Instantaneous  Axis  of  Rotation. — In  Fig,  10,  let  AB 
represent  an  inflexible  and  inextensible  rod,  and  first  let  the 
points  A  and  B  move  in  the  plane  of  the  paper,  in  opposite 
directions  perpendicular  to  AB,  with  velocities  AF,  BK\ 
these  motions  establish  a  rotation  of  the  rod  about  the  point 
/,  determined  by  drawing  FK.  If  we  now  add  a  longi- 
F  D' 


K  EL 

FIG.  10. 

tudinal  component  to  the  motion  of  each  point,  as  AE,  = 
IGy  —  BCj  the  resultant  motions  of  A  and  B  will  become  AD 
and  BH,  and  that  of  7  will  be  IG. 

Draw,  through  A  and  7,  lines  perpendicular  to  AD  and 
IG  respectively,  intersecting  in  O ;  draw  also  DO  and  GO. 
Then  the  triangles  AFD,  AIO,  are  similar,  whence 

AF  _  AD    _FD    _IG 

AP  ~~~  AO'~~  Or  ~  JO' 


VELOCITY  DIAGRAMS.  n 

Therefore  the  angles  AOD,  GOT,  are  equal;  that  is  to  say, 
the  two  points  A  and  /  are  rotating  with  the  same  angular 
velocity  and  in  the  same  direction  about  the  centre  O. 
The  same  must  therefore  be  true  of  every  other  point  of 
the  rigid  bar.  This  may  be  at  once  verified  as  to  the  point 

AD       AF 
B,  as   follows  ;    we    have   already  shown   that  -r-^  —  -jj  ; 

therefore  the  angles  AOD,  AIF,  BIK,  are  equal,  whence 

BK        IG 


_ 

Br~~o~r~   or 

Therefore  the  triangles  BKH,  BIO,  are  similar,  and  BH  is 
perpendicular  to  BO. 

With  reference  to  the  bar  AB,  the  point  O  is  called  the 
instantaneous  axis,  because  during  the  motion  of  the  bar,  it 
changes  its  position  from  instant  to  instant,  not  only  in 
space  but  relatively  to  the  bar  itself  ;  and  it  is  found,  when 
the  simultaneous  motions  of  two  points  of  the  bar  are 
given,  as  for  example  AD,  BH,  by  drawing  through  each 
point  a  line  perpendicular  to  the  direction  of  its  motion  ; 
the  intersection  of  these  perpendiculars  locates  the  axis 
sought. 

And  obviously,  since  the  bar  is  rigid,  any  point  rigidly 
connected  with  it  must  obey  the  same  law,  and  rotate  about 
the  instantaneous  axis  with  the  same  angular  velocity  and 
in  the  same  direction. 

12.  The  Instantaneous  Centre.  —  The  rotation  about  <9,  then, 
is  equivalent  to  a  rotation  about  /,  in  the  same  direction 
and  with  the  same  angular  velocity,  combined  with  a  longi- 
tudinal motion. 

As  a  distinctive  name,  then,  this  point  /  may  properly 
be  called  the  instantaneous  centre. 


12 


VELOCITY  DIAGRAMS. 


It  is,  as  we  have  seen,  the  foot  of  the  perpendicular  let 
fall  from  O  upon  AB.  This  fact  is  sometimes  of  service  in 
locating  the  instantaneous  axis — as,  for  instance,  when  the 
directions  of  the  motions  of  A  and  B  being  assigned,  the 
perpendiculars  to  those  directions,  Ax  and  By,  intersect 
each  other  very  acutely. 

Now,  in  Fig.  10,  FK  cuts  OG  in  L ;  and  in  the  triangles 
AIF,  IOL,  the  angles  API,  OIL,  are  equal  by  reason  of  .the 
parallels  AF,  OI\  and  the  angles  AIF,  IOL,  have  already 
been  shown  to  be  equal.  Therefore,  since  IA  is  perpen- 
dicular to  AF,  IL  is  perpendicular  to  OL;  consequently  OG 
is  perpendicular  to  DH,  which  passes  through  G,  and  is 
parallel  to  FK,  as  was  proved  by  Fig.  8. 

That  is  to  say,  G  is  the  foot  of  the  perpendicular  let  fall 
from  the  instantaneous  axis  upon  the  line  joining  the  ex- 
tremities of  the  resultants. 

13.  Contact  Motions — Normal  and  Tangential  Components. — In 
Fig.  1 1  are  shown  two  pieces  turning  about  the  fixed  cen- 


FlG.    II. 

tres  C  and  D,  and  in  contact  at  P\  if  the  left-hand  one  turn 
as  shown  by  the  arrow,  it  will  push  the  other  out  of  its 


VELOCITY  DIAGRAMS.  13 

way,  and  compel  it  to  turn,  in  this  case  in  the  opposite 
direction.  Draw  the  contact  radii  PC  and  PD\  also  draw 
through  P,  TT  the  common  tangent  to  the  two  curves,  and 
NN  their  common  normal.  The  point  P  of  the  driver  must 
move  in  a  direction  perpendicular  to  PD\  let  its  linear 
velocity  be  represented  by  PA,  which  can  be  resolved  into 
the  normal  component  PE  and  the  tangential  component  PF. 
Of  these  two,  the  latter  is  obviously  non-effective ;  it  repre- 
'sents  merely  the  sliding  of  P  along  the  tangent,  and  has  no 
tendency  to  move  the  follower.  The  motion  of  the  point  P 
of  the  right-hand  piece  must  be  perpendicular  to  PC,  and 
must  have  a  velocity  such  that  its  normal  component  shall 
also  be  PE-— consequently  the  extremity  B  of  this  resultant 
must  lie  in  the  line  AE.  The  component  PH  also  repre- 
sents sliding  along  the  tangent;  it  is  clear,  therefore,  that  in 
this  case  the  actual  sliding  of  one  piece  upon  the  other  will 
bePF—Ptf,  orHF. 

14.  Roberval's  Method  of  Drawing  Tangents.— Whatever  the 
path  described  by  a  moving  point,  the  direction  of  its  mo- 
tion at  any  instant  is  that  of  the  tangent  to  that  path,  at  the 
position  occupied  by  the  point  at  the  given  instant. 

The  law  of  the  motion,  even  in  abstract  mathematics, 
may  often  be  best  explained  by  describing  it  as  dependent 
upon,  and  produced  by,  other  motions:  thus,  for  example,  a 
spiral  of  any  kind  is  usually  defined  as  the  path  of  a  point 
which  travels  along  a  right  line,  while  the  line  itself  revolves 
about  a  fixed  centre. 

And  "  RobervaFs  method  "  of  drawing  a  tangent  to  a  curve 
consists  simply  in  finding  the  resultant  of  these  component 
motions.  A  simple  illustration  of  this  method  is  shown  in 
its  application  to  the  spiral  of  Archimedes,  Fig.  12.  Let  a 
point  move  at  a  uniform  rate  from  P  along  the  right  line  PQ, 


14  VELOCITY  DIAGRAMS. 

while  that  line  revolves  in  the  direction  of  the  arrow  around 
Pas  a  fixed  centre,  also  at  a  uniform  rate.  Let  the  radial 
travel  be  such  that,  while  making  one  revolution,  the  point 
shall  movefrom  Pto^4,  then  it  will  trace  the  curve  PEFAG\ 
and  let  it  be  required  to  draw  a  tangent  at  the  point  O. 
The  two  motions  might  be  supposed  to  take  place  indepen- 
dently ;  if  we  first  imagine  the  rotation  to  be  arrested,  the 


FIG.  12 

point  will  move,  in  the  time  of  one  revolution,  radially  out- 
ward through  a  distance  ON  equal  to  PA  :  and  we  may 
therefore  let  ON  represent  the  velocity  of  this  component. 
If  we  suppose  the  radial  motion  to  be  arrested,  then  in  the 
same  time  the  point  will  describe  the  circumference  of  the 
circle  of  which  PO  is  the  radius.  The  direction  of  this 
motion  at  the  instant  being  that  of  the  tangent  to  the  circu- 
lar path,  we  have  as  the  other  component  OM  perpendicu- 
lar to  PO,  and  equal  in  length  to  the  circumference ;  then, 
completing  the  rectangle,  the  diagonal  OR  is  tangent  at  O 
to  the  spiral,  as  required  :  obviously,  both  components  may 
be  reduced  in  the  same  proportion,  without  affecting  the 
direction  of  the  resultant,  which  is  of  course  independent  of 
the  actual  velocity  of  the  tracing  point. 


VELOCITY  DIAGRAMS.  15 

Since        .  =  tan.  NOR,  the  inclination  of  Jie  tangent  to 


the  radius  vector  can  be  determined  by  elementary  trigo- 
nometry ;  and  in  its  application  to  many  other  curves,  of 
high  and  low  degree,  a  like  simplicity  characterizes  this 
elegant  process.  Nevertheless,  in  his  treatise  on  Descrip- 
tive Geometry,  Mr.  J.  F.  Heather  makes  this  curious  remark  : 
"  This  method,  which  Roberval  invented  before  Descartes 
had  applied  algebra  to  geometry,  is  implicitly  compre- 
hended in  the  processes  of  the  differential  calculus,  on  which 
account  it  is  not  noticed  in  elementary  mathematics  "  ;  — 
where  it  would  seem,  on  the  contrary,  to  deserve  a  conspic- 
uous place  :  it  is  certainly  more  easily  comprehended  than 
the  calculus,  to  which  indeed  it  is  a  natural  prelude.  At 
any  rate,  it  has  a  natural  and  direct  application  to  our  pres- 
ent purpose,  since  in  mechanical  devices  the  actual  motion 
of  a  point  is,  more  often  than  not,  controlled  by  other 
motions  whose  combined  effect  it  is  necessary  to  determine. 

15.  But  in  applying  it,  all  the  circumstances  of  the  case 
must  be  considered,  and  all  the  conditions  which  may  affect 
the  result  must  be  satisfied.  This  suggestion  may  at  first 
glance  appear  superfluous  ;  but  it  seems  otherwise  in  view 
of  the  fact  that  Mr.  Heather,  in  the  only  illustration  of 
Roberval's  method  which  he  gives,  has  conspicuously  failed 
to  act  upon  it.  The  curve  selected  for  this  solitary  exam- 
ple is  the  ellipse,  traced,  as  in  Fig.  13,  by  a  point  P  moving 
in  such  a  manner  as  to  keep  always  taut  a  thread  APB 
attached  by  its  extremities  to  the  foci  A  and  B. 

Mr.  Heather's  explanation  is  as  follows  :  "  Since  the 
length  of  the  string  is  constant,  the  distance  APis  lengthened 
at  each  instant  of  the  motion  by  the  same  distance  as  the 
distance  BP  is  diminished.  The  velocity  of  the  describini_r 


i6 


VELOCITY  DIAGRAMS. 


point  in  the  direction  AP  is  therefore  equal  to  the  velocity 
in  the  direction  PB.  If,  then,  equal  straight  lines  be  cut  off 
from  PB,  and  from  AP  produced,  and  the  parallelogram 
PNOM  be  completed,  the  diagonal  PO  of  this  parallelogram 
will  be  the  direction  of  the  motion  of  the  generating  point 
at  P,  and  consequently  the  tangent  to  the  curve  at  this 


FIG.   13. 

point.  It  is  clearly  seen  from  this,  that  in  the  ellipse,  the 
tangent  bisects  the  angle  BPN  formed  by  one  of  the  focal 
distances  and  the  production  of  the  other,"  etc.,  etc. 

Mr.  Heather  elsewhere  explicitly  states  that  if  the  com- 
ponents in  two  directions  are  given,  whatever  their  relative 
magnitudes,  the  method  of  Roberval  consists  in  "  complet- 
ing the  parallelogram  and  drawing  the  diagonal."  In  the 
above  example  the  direction  of  the  tangent  is,  undeniably, 
correctly  found  in  this  manner — and  the  same  is  true  if  this 
construction  is  applied  to  the  hyperbola,  in  which  the  mov- 
ing point  recedes  from  both  foci  at  the  same  rate,  or  to  the 


VELOCITY  DIAGRAMS.  17 

parabola,  in  which  it  recedes  at  the  same  rate  Irom  the  focus 
and  the  directrix. 

16.  In  all  these  cases,  however,  the  components  are  equal; 
but  the  statement  above  quoted  leads  us  to  expect  a  correct 
determination  whether  they  are  equal  or  not :  let  us  put  it 
to  the  test.  In  Fig.  14,  let  A  and  B  be  two  fixed  points, 
with  reference  to  which  the  point  P  moves  subject  to  the 
condition  that  PB  shall  always  be  twice  PA.  In  other 
words,  the  path  of  P  is  the  locus  of  the  vertices  of  all  tri- 
angles of  which  AB  is  the  common  base,  and  one  of  the  two 
other  sides  is  twice  the  third.  In  order  to  preserve  this 
ratio  it  is  clear  that  P,  in  moving  to  the  right,  must  recede 


FIG.  14. 

from  B  twice  as  fast  as  it  does  from  A.  Consequently,  set- 
ting off  on  the  prolongations  of  BP  and  AP,  PM  =  2PN, 
and  completing  the  parallelogram,  the  diagonal  PO  should, 
in  accordance  with  what  immediately  precedes,  be  the  re- 
quired tangent.  But  it  can  be  shown  that  the  path  in  ques- 
tion is  the  circumference  of  the  circle  whose  diameter  is  DE, 
determined  by  making  BD  =  2AD,  and  BE=.2AE.  It  is 


1 8  VELOCITY  DIAGRAMS. 

obvious  that  PO  is  not  tangent  to  this  circle,  and  therefore 
cannot  be  the  resultant  motion  of  P. 

A  little  reflection  will  show  that  two  important  condi- 
tions have  been  neglected  ;  the  point  P  lies  upon  the  right 
line  PA,  and  must  always  do  so  ;  therefore,  if  that  point 
moves  as  shown,  the  line  must  turn  about  A  as  a  fixed  cen- 
tre ;  and  similar  reasoning  applies  to  the  line  PB.  Thus  the 
situation  is  in  fact  more  complicated  than  it  seemed  at  first, 
and  really  presents  a  special  case  of  the  general  problem  of 
determining  the  motion  of  the  intersection  of  two  right  lines 
rotating  about  fixed  centres  ;  which  therefore  must  next  re- 
ceive attention.  The  elements  of  this  problem  are  embod- 
ied in  the  mechanical  combination  shown  in  Fig.  15,  where 
AC,  BD,  represent  two  steel  rods,  each  formed  into  an  eye 
at  one  end  and  turning  about  the  fixed  pins  C,  D\  upon 
these  rods  two  sleeves  slide  freely,  and  they  are  pivoted 
together  by  a  pin  at  P,  whose  axis,  perpendicular  to  the 
paper,  intersects  the  centre  lines  of  both  rods. 

The  operation  of  this  device  may  perhaps  be  best  studied 
by  first  supposing  one  rod,  as  AC,  to  be  held  stationary, 
while  BD  turns.  The  point  P  of  that  rod  must  then  at  the 
instant  move  in  a  direction  perpendicular  to  BD ;  let  its 
velocity  be  represented  by  PG.  The  pin  connecting  the 
sleeves  must  move  absolutely  in  the  direction  PC,  since  the 
rod  AC  now  forms  a  fixed  guide,  along  which  the  sleeve 
through  which  it  passes  is  compelled  to  slide.  The  other 
sleeve,  however,  not  only  rotates  with  BD,  but  can  slide 
along  that  rod  :  consequently  the  actual  velocity  PI  of  the 
pin  P  is  found  by  drawing  through  G  a  parallel  to  BD. 
Drawing  IH  parallel  to  PG,  we  observe  that  upon  the  sup- 
position above  made  the  pin  P  moves  in  the  direction  AC  at 
the  rate  PI,  and  in  the  direction  BD  at  the  rate  PH. 


VELOCITY  DIAGRAMS.  19 

Next  let  BD  be  held  stationary,  and  let  the  point  Pol  the 
rod  AC  move  in  rotation  about  C  with  the  velocity  PL.  By 
similar  reasoning  we  shall  find  the  resultant  motion  of  the 
pin  P  to  be  PF  in  the  direction  BD,  and  it  will  be  accom- 
panied by  a  motion  PE  in  the  direction  AC. 


FIG.  15 


17.  Now  if  both  rods  rotate  at  once,  with  the  same  veloc- 
ities as  before,  the  final  resultant  motion  PR  of  the  pin  P  is 
found  by  considering  the  partial  resultants  PI,  PF,  as  com- 
ponents, and  completing  the  parallelogram  ;  because,  as  we 
have  just  seen,  these  are  wholly  independent  of  each  other. 
But  it  is  to  be  noted  that  P  will  move  toward  C  with  a 
velocity  equal  to  PI-{-PE,  and  toward  D  with  a  velocity 
equal  to  PF  -\-  PH.  Drawing  RM  perpendicular  to  BD,  and 
RN  perpendicular  to  AC,  we  have  FM=  PH,  and  IN  =  PE  : 
so  that  PM,  PN,  are  the  velocities  of  P  in  the  directions  BD, 
AC,  respectively. 

Also,  since  FR,  IR,  are  merely  prolongations  of  LF  and 
GI,  it  will  be  seen  that  having  assigned  the  components  of 


20  VELOCITY  DIAGRAMS. 


rotation,  P£  and  PL,  the  resultant  PR  may  be  at  once  deter- 
mined by  drawing  perpendiculars  to  them,  which  will 
intersect  in  R  ;  then  drawing  RM  and  RN,  we  determine 
PM  and  PN,  the  total  sliding  components. 

If  then,  as  in  Fig.  16,  the  components  PN,  PM,  are  as- 
signed, the  resultant  is  found,  not  by  completing  the  paral- 
lelogram, but  by  drawing  Nz,  Mw,  respectively,  perpendic- 
ular to  AC  and  BD  ;  these  intersect  in  R,  and  PR  is  the 
resultant.  Had  this  resultant  been  assigned,  the  sliding 
components  are  found  at  once  as  above  stated,  by  reversing 
this  process;  and  the  components  of  rotation  are  determined 
as  readily  by  drawing  Px  perpendicular  to  AC,  and  Py  per- 
pendicular to  BD,  upon  which  lines  we  let  fall  from  R  the 
perpendiculars  RL,  RG. 

18.  This,  then,  is  the  proceeding  which  should  have  been 
adopted  in  Figs.  13  and  14.  Applying  it  in  those  cases,  we 
find  the  resultant  motion  of  the  point  Pupon  the  ellipse  to 
be  PR  instead  of  PO\  the  direction  is  the  same,  but  the 
magnitude  is  different,  and  plainly  will  be  so  except  in  the 
case  when  PA  and  PB  are  perpendicular  to  each  other:  in 
Fig.  13,  when  the  angle  APB  is  obtuse,  as  shown  at  the 
right,  PO  is  too  great  ;  and  when  APB  is  acute,  as  shown  at 
the  left,  PO  is  too  small.  And  in  Fig.  14  it  is  seen  that  the 
direction  as  well  as  the  magnitude  of  the  true  resultant  PR 
is  widely  different  from  that  of  PO  ;  moreover,  it  can  be 
proved  that  PR  is  in  this  case  perpendicular  to  the  radius 
PC,  and  therefore  tangent,  as  it  should  be,  to  the  circular 
path  DPE. 

The  manner  of  determining  the  resultant,  when  the  com- 
ponents PM,  PN,  Fig.  16,  are  assigned,  is  in  this  case  pre- 
cisely  the  same  as  that  employed  in  Fig.  6.  But  the  condi- 
tions are  by  no  means,  the  same  ;  in  that  instance  the  lines 


VELOCITY  DIAGRAMS. 


21 


along  which  the  given  components  lie  intersect  always  in 
the  same  point  A,  whereas  in  this  the  point  of  intersection 
moves  along  both  lines ;  and  they  both  clearly  illustrate  the 
fact  that  Roberval's  method  does  not  always  consist  in 
merely  completing  the  parallelogram  of  two  given  compo- 
nents and  drawing  the  diagonal. 

19.  The  problem  under  consideration  has  thus  far  been 
limited  by  supposing  the  fixed  centres  of  rotation  to  lie 
upon  the  lines  themselves.  It  is  clearly  not  essential  that 
this  should  be  so, — one  or  both  these  centres  may  be  other- 
wise located  ;  and  the  latter  case  is  illustrated  in  Fig.  17, 
where  the  rigid  rods  AS,  BT,  are  riveted  into,  and  form 


O 


FIG.  17. 


FIG.  18. 


parts  of,  the  arms  CS,  TD.  Suppose,  as  before,  one  piece, 
as  ASC,  to  be  held  stationary,  and  let  the  other  move,  the 
point  Pof  the  latter  having  the  velocity  PG ;  the  direction 
of  this  motion  is,  of  course,  perpendicular,  not  to  BT,  but 
to  PD.  The  resultant  motion  of  the  pin  connecting  the 
sleeves  must,  as  before,  be  in  the  direction  PS;  and  in  order 
to  produce  it  there  must  be  compounded  with  PG,  the  com- 
ponent of  rotation,  a  sliding  component  in  the  direction  BT: 


22  VELOCITY  DIAGRAMS. 

because  by  the  structure  of  the  mechanism  these  are  the 
only  motions  of  which  the  sleeve  upon  the  rod  BTis  capa- 
ble. Therefore,  the  first  partial  resultant  PI  is  found  by 
drawing  through  G  a  parallel  to  BT;  and  the  sliding  along 
BT,  by  drawing  through  /a  parallel  to  PG — which  is  per- 
pendicular to  PD. 

Next,  keeping  BTD  stationary,  assign  to  the  point  P  of 
the  other  rod  a  rotative  component  PL,  perpendicular  to 
PC ;  then  by  similar  reasoning  we  find  the  other  partial  re- 
sultant PF,  and  sliding  component  PE.  Compounding 
these  two  partial  resultants  as  in  Fig.  15,  we  obtain  the  final 
resultant  PR,  as  the  motion  of  the  pin  P  when  both  rods 
move  with  the  assigned  velocities; — and  the  total  rate  of 
sliding  along  each  rod  is  found  by  drawing  RN  perpen- 
dicular to  PC,  and  RM  perpendicular  to  PD. 

20.  We  are  now  in  a  position  to  make  the  problem  still 
more  general,  and  to  determine  the  motion  of  the  point  of 
intersection  of  two  curved  lines  rotating  about  fixed  centres. 
In  Fig.  1 8,  the  rigid  steel  rods  ./AT,  QO,  bent  into  the  forms 
of  circular  arcs,  pass  through  correspondingly  curved 
sleeves  pivoted  together  at  P,  and,  being  riveted  into  the 
arms  JD,  QC,  are  compelled  to  rotate  about  the  fixed  cen- 
tres D,  C.  Through  Pdraw  a  tangent  to  each  curve;  for 
facility  of  comparison,  this  combination  has  been  purposely 
so  drawn  that  these  tangents  are  parallel  to  BT,  AS,  of  Fig. 
17,  and  that  the  points  P,  C,  and  D,  have  the  same  relative 
positions  as  in  that  diagram. 

Proceeding  as  before,  we  assign  a  rotative  component 
PG  to  the  point  P,  on  the  arc  JK,  the  right-hand  piece 
being  held  stationary.  Then  it  will  be  at  once  seen  that 
the  first  partial  resultant  PI  must  have  the  direction  of  the 
tangent  PS,  while  the  sliding  component  must  have  the  di~ 


I/ELOCITY  DIAGRAMS.  23 

rection  FT,  tangent  to  JK.  Further  explanation  is  needless, 
it  being  now  quite  apparent  that  the  final  resultant  PR,  and 
the  total  rate  of  sliding,  PM  and  PN,  are  determined  pre- 
cisely as  in  Fig.  17. 

21.  Going  one  step  farther,  it  is  to  be  observed  that  the 
circular  arcs  JK,  OQ,  might  be   tangent  at  P  to  any  two 
curues  whatever,  circular  or  otherwise  ;  and  supposing  those 
curves  to  rotate  about  Cand  D,  the  motion  of  P  at  the  instant 
would  not  be  in  any  way  affected  by  the  substitution.     It  is 
true  that  if  the  new  curves  are  non-circular,  the  device  of 
the  sliding  sleeves  can  not  be  employed.   But  by  this  time  it 
should  have  been  perceived  that  they  are  not  at  all  necessary 
any  case,  and  that  the  motion  of  the  point  of  intersection  of 
the  two  lines,  straight  or  curved,  is  in  fact  determined  by 
the  rotations  alone. 

Nevertheless,  these  sleeves  have,  as  we  think,  been  of 
service  in  the  investigation,  by  making  apparent  to  the  eye 
the  reasons  for  certain  steps  in  the  process  of  finding  the 
resultant  sought.  And  though  just  such  combinations  as 
these  may  not  be  met  with  in  mechanism,  yet  very  similar 
ones  may  be,  and  at  any  rate  it  has  already  been  shown  that 
the  main  question  has  an  important  bearing  on  some  math- 
ematical problems ;  nor  need  we  go  far  to  find  instances  in 
which  its  solution  may  be  of  direct  practical  interest. 

22.  For  example,  given  the  velocities  of  the  blades  of  a 
common  pair  of  shears,  what  is  the  rate  of  shearing  cut  ? 
The  same  question  may  be  asked  in  relation  to  the  pruning 
scissors  with  curved  blades,  shown  in  Fig.  19.     Here  C  is 
the  common  centre  of  rotation,  AS  is  tangent  to  the  upper 
blade,  and  BT  to  the   lower  one,  at  their  intersection  P. 
Regarding    C  as  fixed,  let  PG,  PL,  be  the  components  of 
rotation,   both  of  which  are  perpendicular  to   PC.     Draw 


24 


VELOCITY  DIAGRAMS. 


through  G  a  parallel  to  BT,  and  through  L  a  parallel  to  AS  ; 
these  intersect  in  R,  and  PR  is  the  motion  of  the  point  of 
intersection.  This  will  be  readily  seen  by  the  aid  of  the 
reference  letters  to  be  a  case  precisely  similar  to  that  shown 
in  Fig.  1 8  ;  when  the  blades  are  straight,  as  in  the  common 
shears,  each  edge  is  its  own  tangent,  and  the  construction  is 
made  as  in  Fig.  17. 


FIG.  19. 

23-  The  direction  and  velocity  of  the  motion  of  a  point 
may  also  be  determined  in  many  cases  by  means  of  the  in- 
stantaneous axis.  For  example :  In  Fig.  20,  C  and  D  are  the 
fixed  centres  upon  which  turn  the  levers  CA,DB  ;  the  points 
A  and  B  are  connected  by  the  link  AB,  which  is  extended 
in  a  curved  form  to  P:  the  motion  of  A  is  necessarily  per- 
pendicular to  CA,  and  letting  AM  represent  its  velocity,  it 
if  required  to  find  the  velocity  of  the  points  B  and  P.  Since 
B  is  constrained  to  move  in  a  direction  perpendicular  to  DB, 
the  instantaneous  axis  is  found  by  producing  CA,  DB,  until 
they  intersect  in  O ;  then  drawing  OM,  the  angle  represents 


VELOCITY  DIAGRAMS.  2$ 

the  angular  velocity  of  A  around  O :  and  this  must  be  the 
same  for  every  point  of  the  rigid  piece  ABP.  Also,  the 
direction  of  the  motion  of  every  such  point  must  be  perpen- 
dicular to  the  instantaneous  radius,  or  line  drawn  from  the 
point  in  question  to  the  instantaneous  axis.  Therefore, 
drawing  through  B  a  perpendicular  to  BO,  and  through  Pa 
perpendicular  to  PO,  the  required  velocities  BN,  PR,  are 
determined  by  making  the  angles  6f,  d" ',  each  equal  to  tf,  or 
AOM. 

In  the  present  instance  this  affords,  perhaps,  the  most 

/O 


FIG.  20. 

simple  means  of  determining  the  motion  of  P ;  arid  since  the 
direction  of  this  motion  must  be  tangent  to  the  path  of  that 
point,  attention  is  thus  drawn  to  the  fact  that  by  the  use  of 
the  instantaneous  axis,  a  tangent  to  a  curve  may  sometimes 
be  drawn  even  more  readily  than  by  Roberval's  method. 

24.  But  the  circumstances  may  be  such  that  the  instan- 
taneous axis  is  inaccessible,  or  that  its  determination  is  un- 


36  VELOCITY  DIAGRAMS. 

reliable  ;  one  or  both  of  which  things  would  have  resulted  if, 
in  Fig.  20,  the  two  levers  had  been  nearly  parallel.  In  this 
event  other  means  must  be  employed,  and  in  Fig.  21  a  dif- 
ferent process  is  illustrated,  the  conditions,  for  facility  of 
comparison,  being  the  same  as  in  the  preceding  figure.  The 
assigned  motion  AM  has  an  absolute  component  Ae  in  the 
direction  AB,  and  the  motion  of  B  must  have  an  equal  com- 
ponent Be'  in  the  same  direction.  The  actual  motion  of  B 
must  be  perpendicular  to  BD,  and  its  velocity  £Aris  deter- 
mined by  drawing  through  e'  a  perpendicular  to  Ae' .  Draw 
BP ;  then  BN  will  have  an  absolute  component  Bg  in  this 
direction,  to  which  Pg*  must  be  equal.  Also  draw  AP\ 


then  AM  has  an  absolute  component  Af  in  that  direction,  to 
which  Pf  is  equal.  Through  g'  draw  a  perpendicular  to 
Bg',  and  through/'  a  perpendicular  to  Af ;  these  intersect 
in  R,  and  PR  is  the  required  motion  of  the  point  P. 

25.  Composition  of  Revolution  and  Rotation. —  Numerous 
mechanical  devices  are  practically  employed,  in  which  one 
wheel  not  only  rotates  about  its  own  axis,  but  at  the  same 


VELOCITY  DIAGRAMS.  27 

time  travels  in  an  orbit  about  another  fixed  axis,  thus  con- 
stituting what  is  called  a  planetary  train.  Any  point  con- 
nected with  the  planetary  wheel  then  travels  in  a  path  de- 
pendent upon  both  these  circular  motions,  and  consequently 
called  an  aggregate  path.  The  general  principle  is  illustrated 
in  Fig.  22  ;  the  wheel  W,  whose  centre  is  C,  has  its  bearings 
in  the  train-arm  AD,  which  turns  about  D  as  a  fixed  centre ; 
and  this  wheel  may  be  made  to  turn  in  its  bearings,  inde- 
pendently of  the  motion  of  the  arm,  by  any  suitable  means. 
Supposing  the  arm  to  be  stationary,  let  the  wheel  rotate  as 
shown  by  the  arrow  x,  with  such  a  rate  that  a  pin  P  fixed  in 
the  side  of  W  shall  move  with  a  velocity  PQ,  the  direction 
of  this  motion  being  necessarily  perpendicular  to  PC. 

Next,  suppose  PFnot  to  turn  at  all  about  C,  and  let  the 
train-arm  move  as  shown  by  the  arrow  j,  the  velocity  of  A 


R 


M 


FIG.  22. 


being  represented  by  AM,  perpendicular  to  AD.  Since  the 
arm  and  the  wheel  now  move  as  one  piece,  this  will  impart 
to  P  a  motion  perpendicular  to  PD,  the  velocity  PN  being 


28  VELOCITY  DIAGRAMS. 

determined  by  making  the  angle  $'•  equal  to  the  angle  d. 
Now,  if  these  two  movements  take  place  simultaneously,  the 
resultant  motion  of  P  will  be  determined  by  compounding 
PQ  and  /W, — that  is,  by  completing  the  parallelogram  and 
drawing  the  diagonal  PR;  because  the  rotation  about  C  and 
the  revolution  about  D  are  wholly  independent  of  each 
other,  both  in  direction  and  in  velocity. 

26.  Before  proceeding  to  the  construction  of  a  complete 
velocity  diagram,  we  propose  to  give  a  series  of  examples 
showing  how  the  principles  and  processes  above  described 
may  be  combined  and  applied  in  determining  the  motions 
of  certain  points  in  detached  mechanical  movements.     It  is 
immaterial   whether  the  combinations  selected  are,   or  are 
not,  parts  of  actual  machines,  because  it  would  be  difficult 
to    contrive   a  "  mechanical  movement  "  of  practical  form 
which  might  not  some  time  be  found  adaptable  to  a  useful 
purpose.     And  the  study,  or  better  still,  the  actual  execu- 
tion, of  such  exercises,  in  which  attention  is  confined  to  the 
action  of  comparatively  few  parts,  will  be  found  the  best 
means  of  acquiring  thorough  familiarity  with  the  principles, 
and  facility  in  their  application. 

27.  In  the  first  of  these  examples,  Fig.  23,  the  lever  CP  is 
pivoted  at  C  to  a  socket  which  slides  along  the  fixed  guide 
TT,  and  at  A  the  link  AQ  is  joined  to  this  lever.     Let  AL 
be  the  absolute  component  of  the  motion  of  A,  in  the  line  of 
this  link,  and  let  CM  be  the  resultant  motion  of  C  along  the 
guide  ;  it  is  required  to  find  the  actual  motions  of  the  points 
A  and  P.     Resolve  CM  into  the  components  CD  in  the  di- 
rection PC,  and  CE  perpendicular  to  it ;  then  PH,  AF,  will 
be  the  absolute  components  of  the  motions  of  Pand  A  in  the 
line  of  the  lever,  each  being  equal  to  CD.     Draw  through  L 
a  perpendicular  to  AQ,  and  through  F  a  perpendicular  to 


VELOCITY  DIAGRAMS. 


29 


* 

6 

£ 


30  VELOCITY  DIAGRAMS. 

PC;  these  intersect  in  N,  and  AN  is  the  required  resultant 
motion  of  A.  The  side  component  of  AN  is  AG ;  and  the 
two  side  components  AG,  CE,  determine  a  rotation  of  PC 
about  the  point/,  where  it  intersects  the  prolongation  of  GE. 
The  prolongation  of  EG  limits  PK,  the  side  component  of 
the  motion  of  P,  which,  compounded  with  PH,  determines 
PR,  the  required  resultant. 

One  test  of  the  accuracy  of  the  construction  is  to  draw 
the  line  RNM  through  the  extremities  of  the  resultants  ; 
this  should  be  parallel  to  KGE,  and  when  produced  to  cut 
the  prolongation  of  PC  in  B,  IB  should  be  equal  to  CD. 

Another  test  is  to  find  the  instantaneous  axis  of  CP,  by 
drawing,  through  P  and  C,  lines  perpendicular  to  PR  and 
CM  respectively.  These  intersect  in  O  :  and  not  only  should 
all  the  angles  marked  #  be  equal  to  each  other,  but  (9/should 
be  perpendicular  to  PI,  and  OB  perpendicular  to  RB. 

28.  In  Fig.  24,  a  lever  turning  about  the  fixed  centre  D 


R 


FIG.  24. 

is  jointed  at  P  to  a  rod  PR,  which  slides  freely  through  a 
sleeve  formed  in  the  outer  end  of  a  rigid  arm  EC,  which 


VELOCITY  DIAGRAMS.  31 

turns  about  the  fixed  centre  C.  Given  the  motion  PA  of  the 
point  P,  it  is  required  to  find  the  motions  of  R  and  of  the 
point  E  of  the  arm  EC,  and  also  the  rate  of  sliding  of  the 
rod  through  the  sleeve. 

The  only  motions  of  which  the  rod  PR  is  capable,  are 
those  of  revolving  around  the  centre  C,  and  of  sliding  end- 
wise through  the  sleeve.  We  may,  then,  resolve  PA  into 
the  sliding  component  PN  in  the  line  of  the  rod,  and  the 
rotative  component  PM  perpendicular  to  PC.  The  former 
gives  at  once  the  rate  of  sliding,  and  the  latter  determines 
the  angle  PCM,  or  /?,  which  represents  the  angular  velocity 
of  the  rotation  about  C ;  this  is  the  only  motion  of  which 
the  curved  arm  is  capable,  and  EF,  the  required  motion  of 
the  point  E  on  that  arm,  is  perpendicular  to  EC,  and  limited 
by  making  the  angle  ECF  equal  to  p.  The  motion  of  R  has 
also  a  rotative  component  RH,  determined  in  like  manner, 
and  also  a  longitudinal  component  RG,  =  PN;  completing 
the  parallelogram,  RB  is  the  resultant  sought. 

Otherwise,  the  Instantaneous  axis  of  the  rod  PR  might 
have  been  found  thus  :  draw  CI  perpendicular  to  PR,  then, 
in  rotation  about  C,  the  motion  of  the  point  7  would  be  per- 
pendicular to  CI,  and  therefore  longitudinal  in  respect 
to  the  rod.  This  must  be  true  also  in  the  rotation  about  the 
instantaneous  axis  of  the  foot  of  the  perpendicular  from  that 
point  upon  the  rod ;  consequently,  that  foot  can  be  no  other 
than  the  point  /,  and  the  instantaneous  axis  O  may  thus  be 
found  by  producing  CI  to  cut  the  prolongation  of  DP. 
Then  drawing  AO  and  OR  we  may  determine  the  absolute 
motions  of  R  and  of  /  by  making  the  angles  marked  d  equal 
to  each  other. 

A  test  of  this  may  be  applied  by  resolving  PA  into 
the  components  PQ  along  the  rod,  and  PS  perpendicular 


32  VELOCITY  DIAGRAMS. 

to  it ;  resolve  RB  in  a  similar  manner,  then  PQ,  IW,  and  RU, 
should  be  equal  to  each  other,  and  52"  should  pass  through  7. 

The  point  E  of  the  rod  must  have  an  absolute  component 
EV  =  PQ',  this  does  not  indicate  the  whole  sliding,  since 
EF,  the  motion  of  E  on  the  arm,  has  a  sliding  component  EJ 
in  the  opposite  direction  ;  the  sum  of  these,  or  JV,  is  equal 
to  PN,  which  was  previously  determined  by  another  method. 

29.  In  Fig.  25,  D  is  the  fixed  centre  about  which  turns 
the  lever  DB,  jointed  at  B  to  the  bar  ABP.  The  motion  of 
A  is  constrained  by  the  lever  AC,  which  turns  about  C  as  a 
fixed  centre.  Given  the  motion  BM,  it  is  required  to  find 
the  motion  of  P. 

This  may  be  done  in  three  different  ways. 

First.  Resolve  BM  into  the  rectangular  components  BE, 
BF,  then  AF' ,  PF" ,  must  be  equal  to  ^F  and  in  the  same  di- 
rection. The  motion  of  A  must  be  perpendicular  to  AC,  is 
limited  by  F'N  perpendicular  to  AB,  and  has  the  side  com- 
ponent AE'.  A  line  through  E'  and  E  limits  the  side  com- 
ponent PE",and  PR  is  found  by  compounding  PE"  with  PF" . 

Second.  (Same  Figure.)  Produce  CA  to  cut  BD  in  O, 
the  instantaneous  axis  of  AP ;  draw  MO  and  PO,  also  Px 
perpendicular  to  PO.  Construct  the  angle  POy  equal  to  the 
angle  BOM  \  Px  cuts  Oy  in  R,  and  PR  is  the  resultant 
sought. 

Third.  In  Fig.  26,  find  BF,  AN,  and  AE' ,  as  in  the  first 
solution.  Now,  the  motion  of  P  may  be  regarded  as  com- 
pounded of  a  rotation  around  B  and  a  revolution  around  D. 
The  former  component  is  PG  perpendicular  to  ABP,  limited 
by  the  prolongation  of  E' B  ;  the  latter  is  PH  perpendicular 
to  PDj  determined  by  making  the  angle  PDH  equal  to  the 
angle  BDM.  Completing  the  parallelogram,  the  diagonal 
PR  is  the  required  resultant. 


VELOCITY  DIAGRAMS. 


33 


FIG.  26. 


34  VELOCITY  DIAGRAMS. 

Either  of  these  processes  may  be  used  as  a  check  upon 
the  other — and  in  practice  it  is  often  advisable  to  apply  such 
tests,  which  may  lead  to  the  detection  of  an  error ;  or,  what 
is  still  more  satisfactory,  prove  that  none  has  been  made. 

30.  Of  Rolling  Contact. — There  are  numerous  mechanical 
devices  in  practical  use,  in  which  the  motions  of  certain 
parts  are  determined  by  their  connection  with  a  curve  which 
rolls  upon  another  line  which  is  itself  stationary,  and  may 
be  either  curved  or  straight. 

The  nature  of  perfect  rolling-  contact  may  perhaps  be 
best  illustrated  by  a  study  of  that  which  is  not  perfect. 
Thus,  in  Fig.  27,  the  polygon  rolls  upon  the  fixed  right  line 
with  a  hobbling  motion  ;  in  the  position  shown,  the  point  A 
is  at  rest,  and  the  whole  figure  turns  about  it  as  a  centre 
until  B  comes  into  LM  at  D ;  it  will  then  turn  about  B  until 
the  face  BE  coincides  with  LM,  and  so  on  :  the  perimeter 
of  the  polygon  "  measuring  itself  off  "  upon  the  straight  line 
along  which  it  rolls. 

If  the  number  of  sides  be  increased,  their  length  will  be 
less,  and  the  hobbling  will  be  diminished,  until,  when  the 
number  becomes  inconceivable,  it  will  be  come  imperceptible. 
The  broken  contour  then  becomes  the  dotted  curve,  tangent 
to  the  base  line  upon  which  it  rolls,  and  the  change  from 
one  centre  of  rotation  to  another  goes  on  continuously. 
But  this  does  not  alter  the  facts,  that  at  any  instant  the 
point  of  contact  is  at  rest,  and  that  every  point  of  the  curve, 
as  well  as  every  point  rigidly  connected  with  it,  is  at  the 
instant  turning  about  that  point  of»contact  as  a  fixed  centre. 

In  this  case  the  polygon,  being  a  regular  one,  ultimately 
becomes  a  circle ;  and  when  it  does,  the  path  of  its  centre  is 
a  right  line  parallel  to  LM.  And  it  will  now  be  apparent 
that  the  rolling  of  this  circle  upon  is  tangent  is  the  resultant 


VELOCITY  DIAGRAMS.  35 

of  a  rectilinear  translation  indicated  by  the  arrow  /,  and  a 
rotation  about  the  centre  C  of  the  circle  indicated  by  the 
arrow  r.  On  LM  set  off  AN  equal  to  the  quarter  circumfer- 
ence ABE,  and  erect  the  perpendicular  NC ;  then  if  the 
circle,  without  rotating,  be  moved  bodily  to  the  right  until 
C  reaches  C' ,  the  single  point  A  of  the  circle  will  have  been 
brought  into  contact  with  every  point  of  AN, — and  this  is 
pure  sliding  contact.  On  the  other  hand,  we  may  suppose 
C  to  remain  fixed,  and  the  circle  to  turn  as  shown  by  the 
arrow  r  through  the  angle  EGA,  or  90°;  then  every  point  of 
the  quadrant  ABE  will  have  been  brought  into  contact  with 
the  single  point  A  of  the  tangent, — and  this  also  is  pure 
sliding  contact. 

31.  But  if  these  two  motions  take  place  simultaneously 
and  uniformly,  it  is  obvious  that  the  circumference  will 
measure  itself  off  upon  the  tangent,  the  point  I  on  the  arc 
going  to  i '  on  the  tangent,  2  going  to  2',  and  E  going  to  N. 
The  consecutive  points  of  the  arc,  then,  come  into  coinci- 
dence with  the  consecutive  points  of  the  line,  each  in  their 
order  of  sequence ;  and  this  is  pure  rolling  contact ;  no  point 
of  either  comes  in  contact  with  more  than  one  point  of  the 
other,  and  the  length  of  the  line  rolled  over  is  precisely 
equal  to  the  length  of  the  arc  which  rolls  over  it.  More- 
over, it  is  clear  that  the  motions  of  the  point  A,  due  to  the 
rotation  r  and  the  translation  /,  respectively,  are  opposite 
in  direction  and  equal  in  velocity  ;  they  therefore  neutralize 
each  other,  leaving  the  point  of  contact  for  the  instant  sta- 
tionary, as  previously  stated. 

Now,  in  Fig.  28,  the  base  line  LM,  instead  of  being 
straight,  is  a  part  of 'a  circle  whose  centre  is  D.  Set  off  the 
arc' AN  equal  to  the  quadrant  AE ;  then  if  the  point  A  of  the 
upper  circle  slides  over  the  arc  AN,  since  the  two  circles 


VELOCITY  DIAGRAMS. 


must  be  always  tangent,  both  C  and  A  must  move  with  the 
same  angular  velocity  about  D.  This  being  true  of  two 
points  in  the  moving  circle,  must  be  true  of  all  other  points ; 
the  movement  above  supposed  would  therefore  bring  C  to 


FIG.  27. 


FIG.  28. 


C,  and  E  to  .£',— the  arcs  AN,CC,  EE ',  all  having  the  com- 
mon  centre  D,  and  measuring  equal  angles. 

32.  That  is  to  say,  the  motion  represented  by  the  arrow 
t  is  one  of  revolution  about  D,  the  centre  of  curvature  of  the 


VELOCITY  DIAGRAMS. 


37 


base  line  LM—  which  in  Fig.  27  is  one  of  straight  translation 
simply  because  the  radius  of  curvature  of  LM  is,  in  that 
case,  infinite. 

In  either  case,  the  linear  velocity  of  A,  due  to  the  rota- 
tion about  C,  indicated  by  r,  is  equal  and  opposite  to  that 
due  to  the  revolution  indicated  by  /  ;  the  point  of  contact 
being  therefore  at  rest  for  the  instant. 

This  fact  of  itself  seems  almost  conclusive  proof  that  the 


FIG.  29. 

point  in  question  is  the  instantaneous  axis  of  the  moving 
piece  :  this,  however,  is  capable  of  still  more  general  and 
rigorous  demonstration,  which  we  give  here,  because  the 
fact  is  of  importance  in  relation  to  some  combinations  pres- 
ently to  be  considered,  and  ought  therefore  to  be  firmly 
established. 


38  VELOCITY  DIAGRAMS. 

33.  In  Fig.  29,  neither  the  fixed  nor  the  moving  curve  is 
circular,  but  C  is  the  centre  of  curvature  of  the  lower  one, 
and  D  that  of  the  upper  one,  at  their  common  point  A  ; 
therefore  CD  is  their  common  normal.  Let  o,  s,  represent 
points  on  these  curves  consecutive  to,  and  equidistant  from, 
A.  Then,  if  the  upper  curve  slide  upon  the  lower,  the  point 
o  can  be  brought  into  coincidence  with  A  of  the  lower  curve 
only  by  a  rotation  about  D,  indicated  by  the  arrow  r;  and 
the  point  A  of  the  upper  curve  can  be  brought  into  co- 
incidence with  s  only  by  a  revolution  about  C,  indicated  by 
the  arrow  t.  Let  the  linear  velocities  of  the  point  A  of  the 
upper  curve,  due  to  these  motions,  be  represented  by  Ax, 
Az,  and  let  these  be  equal  to  each  other,  since  Ao,  As,  are 
equal  by  hypothesis, 

Now  let  P  be  any  point  rigidly  connected  to  the  moving 
curve.  The  motion  of  P  will  be  the  resultant  of  rotations 
about  D  and  C ';  the  components  are  therefore  perpendic- 
ular to  PD  and  PC,  respectively,  and  as  their  values  we  shall 
have 

A^.PD    1 

AD  PH       PD.  AC 

A*.  PC    I         PE^PC.AD* 


Completing  the  parallelogram,  the  diagonal  PR  is  the 
resultant  motion  of  P.  Next  draw  PA,  and  also  AN  parallel 
to  PC,  and  AK  parallel  to  PD.  We  shall  then  have  : 

PN      AC      .    pN==PD.AC    i 


PN       PD.AC 

and  K-  pK-pc.AD'     (2)' 

PK_AD     .    pK       PC.  AD 
PC  ""  CD'  *  '  CD 


VELOCITY  DIAGRAMS.  39 

The  second  members  of  Equations  (i)  and  (2)  being  iden- 

PH       PN 

tical,  their  first  members  are  equal ;  i.e.,  — —  =  — — . 

P  K        PK. 

But  the  angles,  HPE,  NPK,  are  equal ;  therefore  the 
parallelograms  HE,  NK,  are  similar,  and  the  angles  RPE, 
A  PC,  are  equal.  Consequently, 

EPA  +  RPE,  =  RPA,  is  equal  to  EPA  +  A  PC,  =  EPC. 
Now  EPC  is  a  right  angle  by  hypothesis  ;  therefore  PR, 
the  absolute  motion  of  P,  is  perpendicular  to  a  right  line 
drawn  from  P  to  the  point  of  tangency  A:  Since  P  was 
chosen  at  pleasure,  it  follows  that  lines  perpendicular  to  the 
motions  of  any  other  points  connected  rigidly  with  the 
moving  curve,  will  intersect  in  A,  which  must  therefore  be 
the  instantaneous  axis  of  that  curve  :  and  it  is  perfectly 
clear  that  a  rotation  about  A  will  bring  the  point  o  into 
coincidence  with  s,  which  we  have  already  seen  to  be  a  con- 
dition of  rolling  contact. 

34.  The  utility  of  this  fact  is  well  illustrated  in  Fig.  30, 
which  shows  a  simple  planetary  train.  The  shaft  of  the 
wheel  W,  which  rolls  around  the  fixed  wheel  W,  has  its 
bearings  in  the  crank  or  "  train-arm  r'  CD,  and,  projecting 
through  that  arm,  has  secured  upon  it  a  second  crank  DA. 
The  crank-pin  A  operates  the  link  AP,  of  which  the  opposite 
end  Pis  caused  by  guides,  not  shown,  to  travel  in  the  line 
CP.  The  point  A  traces  the  epitrochoid  shown  in  dotted 
line ;  and  its  position,  for  any  given  position  of  CD,  is  read- 
ily found  as  follows  :  starting  with  the  two  cranks  coinciding 
in  one  right  line  Cda,  and  m  as  the  point  of  contact,  set  off, 
on  the  two  circles,  equal  arcs  mO,  mo ;  then  in  the  rolling,  o 
will  go  to  O,  0</then  becoming  the  contact  radius  OD,  while 
the  angle  oda  will  remain  unchanged. 

The  motion  of  A,  for  any  assigned  velocity  of  D,  might 


VELOCITY  DIAGRAMS. 


\ 


VELOCITY  DIAGRAMS.  41 

be  determined  as  in  Fig.  22,  since  it  is  the  resultant  of  a 
revolution  around  C  and  a  rotation  about  D ;  these  are 
essentially  independent  of  each  other,  notwithstanding  the 
engagement  of  th,e  two  wheels,  which  is  but  one  of  many 
possible  methods  of  fixing  the  relative  directions  and  veloci- 
ties of  those  two  motions. 

But  the  determination  can  be  made  in  a  much  more 
simple  and  direct  manner,  because  O  is  the  instantaneous 
axis.  Let  DN  be  the  motion  assigned  to  D  ;  then  the  motion 
of  A  must  be  perpendicular  to  the  instantaneous  radius  OA, 
and  its  velocity  A M  is  found  by  making  the  angle  AOM 
equal  to  the  angle  DON. 

If  it  be  further  required  to  find  the  velocity  of  P,  the 
argument  is,  that  on  the  prolongation  of  AP  a  component 
should  be  set  off  equal  to  AL  the  absolute  component  of 
AM,  and  at  its  extremity  a  perpendicular  to  AP  should  be 
erected,  which  would  cut  CP  produced,  at  a  point  limiting 
the  required  resultant.  This,  obviously,  would  result  in  the 
construction  of  a  right-angled  triangle  similar  and  equal  to 
ALR,  formed  by  drawing  through  A  a  parallel  to  CP,  cut- 
ting ML  (produced  if  necessary)  in  R ;  which  gives  AR  as 
the  required  velocity  of  P.  This  abbreviation  may  evidently 
be  employed  in  any  case  where  the  motion  of  one  end  of  a 
link  is  given,  and  that  of  the  opposite  end  in  a  given  direc- 
tion is  required. 

35.  Fig.  31  also  exhibits  a  simple  planetary  combination, 
in  which,  however,  the  central  (or  sun)  wheel  IV  is  not"  fixed, 
but  turns  freely  about  the  axis  C.  The  planet-wheel  W  is 
rigidly  secured  to  the  connecting-rod  LP,  the  point  P  being 
made  to  travel  in  the  line  CP;  and  the  two  wheels  are  kept 
in  gear  by  means  of  the  link  CD.  This  arrangement  is 
known  as  "  Watt's  sun-arid -planet  wheels,"  having  been  em- 


VELOCITY  DIAGRAMS. 


VELOCITY  DIAGRAMS.  43 

ployed  by  the  illustrious  engineer  as  a  substitute  for  the 
crank,  upon  which,  as  connected  with  a  rotative  steam 
engine,  some  one  had  secured  a  patent. 

If,  now,  we  assign  to  D  a  velocity  DN.  the  motion  of  P  is 
ascertained,  as  usual,  by  making  PQ  equal  to  DB,  the  abso- 
lute longitudinal  component  of  DN,  and  drawing  QR  per- 
pendicular to  DP.  It  is  next  to  be  noted  that  the  motion  of 
W  consists  of  a  revolution  about  C  and  a  rotation  about  its 
own  centre  D\  and  the  same  is  true  of  the  motion  of  P,  since 
it  is  rigidly  connected  with  that  wheel.  Resolving  PR 
accordingly,  PE  is  the  component  of  revolution,  and  FF 
that  of  rotation.  The  angular  velocity  of  the  revolution  is 
represented  by  the  angle  PCE,  which  of  course  is  equal  to 
DCN\  and  AJis  the  linear  velocity  oi  A,  due  to  this  motion. 
The  angular  velocity  of  the  rotation  is  represented  by  the 
angle  PDF,  and  Lfis  the  additional  circumferential  velocity 
due  to  that  motion,  to  which  JM  is  made  equal,  giving  AM 
as  the  actual  velocity  of  A:  and  ACM  represents  the 
angular  velocity  of  W. 

36.  A  test  of  the  accuracy  of  both  the  analysis  and  the 
construction  may  be  applied  by  producing  DC  to  cut  the 
prolongation  of  EP  in  (9,  the  instantaneous  axis  of  the  planet- 
wheel  and  its  attached  rod.  Draw  NO  and  RO\  then  the 
two  angles  marked  /3  should  be  equal,  and  NO  should  pass 
through  the  point  M\  also  it  is  to  be  observed  that  when  O 
is  accessible,  this  affords  the  readiest  means  of  determining 
A M  when  DN  is  given,  and  vice  versa.  * 

It  is  quite  apparent  that  if  the  velocity  of  D  is  constant, 
the  velocity  of  W  will  vary,  unless  DP  is  infinite,  W  then 
having  a  motion  of  circular  translation  ;  on  the  other  hand, 
if  W  turns  uniformly,  the  motion  of  D  will  be  variable. 
Consequently  the  law  of  variation  in  the  piston  speed  will 


44  VELOCITY  DIAGRAMS. 

be  different  in  an  engine  in  which  this  arrangement  is  used, 
from  that  obtaining  in  one  provided  with  an  ordinary  crank, 
the  main  shaft  revolving  uniformly  in  each  case.  If  the 
circumferential  velocity  of  W  be  assigned,  as  AM,  the 
velocity  of  P  may  be  found  thus :  draw  PA  and  produce 
it,  find  the  absolute  component  AK  along  that  line,  make 
PH  equal  to  it,  and  draw  HR  perpendicular  to  PA  ;  other- 
wise find  the  instantaneous  axis  O,  and  make  the  angle 
FOR  equal  to  the  angle  AOM. 

37.  In  Fig.  32,  the  engaging  wheels  of  which  W,  W,  are 
the  pitch  circles,  turn  about  the  fixed  centres  D  and  C.  In 
the  front  face  of  W  is  fixed  a  pin  B,  turning  freely  in  a 
block,  which  slides  in  a  slot  formed  in  one  arm  of  a  bent 
lever ;  this  lever  turns  about  the  fixed  centre  E,  and  in  the 
extremity  F  of  the  other  arm  is  a  pin  upon  which  is  hung 
a  swinging  lever  FP\  from  the  front  face  of  /^projects  a 
pin  Ay  connected  with  the  free  end  of  the  swinging  lever  by 
a  link  AP.  Considering  A  as  the  driving  point,  and  assign- 
ing t'o  it  a  motion  AM  (necessarily  perpendicular  to  the 
radius  DA),  it  is  required  to  determine  the  motion  of  P. 

Produce  DA  to/  on  the  circumference  of  W,  at  which 
point  draw  a  tangent,  limited  by  its  intersection  at  L  with 
the  prolongation  of  DM  -,  JL  thus  determined  is  the  circum- 
ferential velocity  of  W.  That  of  W  is  necessarily  the  same  ; 
therefore,  producing  CB  to  cut  the  circumference  at  Jf, 
draw  the  tangent  J'L'  equal  to  JL ;  draw  CL' ,  and  at  B 
draw  a  perpendicular  to  CB,  cutting  CL'  in  N\  then  BN  is 
the  linear  motion  of  B.  Resolve  BN  into  two  components, 
one  in  the  line  BE,  the  other  perpendicular  to  it;  the  longi- 
tudinal component  merely  produces  a  sliding  of  the  block 
in  the  slot,  but  the  side  component  Bd  establishes  a  rotation 
of  the  bent  lever  about  E,  of  which  the  angular  velocity  is 


VELOCITY  DIAGRAMS. 


45 


O» 


46  VELOCITY  DIAGRAMS. 

represented  by  the  angle  /?,  or  BEd.  The  point  F  must 
have  the  same  angular  velocity,  and  must  move  in  a  direc- 
tion perpendicular  to  EF9  so  that  its  linear  velocity  FQ  is 
determined  by  constructing  the  angle  ft'  equal  to  /?,  and  the 
absolute  component  of  FQ  along  PF  is  Fe,  to  which  Pe'  must 
be  equal. 

Returning  now  to  the  pin  A  ;  its  motion  AM  has  an  ab- 
solute component  Aa  in  the  line  AP,  to  which  Pa'  must  be 
equal.  Then,  erecting  at  a'  a  perpendicular  to  AP,  and  at 
e  a  perpendicular  to  FPy  the  intersection  R  of  these  per- 
pendiculars is  the  extremity  of  the  required  resultant  mo- 
tion PR. 

38.  In  Fig.  33,  the  planet-wheel  W,  which  rolls  inside 
the  fixed  annular  sun-wheel  W,  has  its  bearings  at  the  ex- 
tremity C  of  one  arm  of  a  bent  lever  CDE,  which  turns 
about  D  the  centre  of  W.  At  the  extremity  E  of  the  other 
arm  of  this  lever,  is  the  bearing  of  a  pin  which  projects  from 
the  farther  side  of  the  arm  or  lever  £F,  and  is  made  in  one 
piece  with  it.  The  lower  end  of  this  arm  is  formed  into  a 
sleeve,  through  which  slides  freely  the  rod  J5S,  perpen- 
dicular to  EF ';  this  rod  is  formed,  at  the  left-hand  end,  into 
an  eye,  fitted  upon  a  pin  B,  which  is  fixed  in  the  planet- 
wheel  and  projects  from  its  front  face.  Finally,  to  the  free 
end  of  EFis  pivoted  the  link  or  connecting-rod  FP,  of  which 
the  farther  end  P  is  made  to  travel  in  the  line  PD. 

Suppose  for  the  moment  the  annular  wheel  W  to  be 
removed,  and  the  bent  lever  CDE  to  be  stationary,  while 
W  revolves  about  C  as  a  fixed  centre.  Then  it  is  to  be 
noted  that  the  kinematic  action  of  the  virtual  crank-arm  CB 
and  the  bent  lever  BEF'is  precisely  the  same  in  this  figure 
as  in  the  preceding  one ;  in  this  case  the  sliding  in  the 
direction  BE  occurs  in  the  sleeve  at  £,  whereas  in  Fig.  32  it 


VELOCITY  DIAGRAMS 


47 


48  VELOCITY  DIAGRAMS. 

was  accommodated  by  the  motion  of  the  block  at  B  in  the 
slotted  ahn  BE.  In  either  case,  then,  the  rotation  of  /> 
about  C  causes  Fto  vibrate  in  a  limited  arc  about  E.  Now 
when  W  is  replaced  and  CDE  is  made  to  revolve  about  D, 
this  vibration  goes  on  as  before,  so  that  the  length  of  the 
virtual  crank-arm  DF,  by  which  P  is  actuated,  is  continually 
varying  within  certain  limits.  Let  C  turn  about  />as  indi- 
cated by  the  arrow  t,  then  W  will  rotate  about  C  in  the 
direction  shown  by  the  am>w  r. 

39.  Assign  to  Ca  velocity  CM,  then  its  angular  velocity 
about  the  instantaneous  axis  A  will  be  represented  by  the 
angle  CAM\  the  motion  of  /.'will  In-  |>erpenilirnlar  to  the 
instantaneous  radius  AB,  and  its  linear  velocity  TWis  deter- 
mined  by  making   the  angle  BAN  equal  to  CAM.     The 
motion  of    /:'  is  /•'/     j.ci  pemliciilar   to    />/•',  the  anisic    /•'/>/ 
being  made  equal  to  the  angle  CDM.    The  absolute  com- 
poncnts  of  BN  are  Bb,  Bn ;  those  of  EL  are  Ee,  Eh.    Join 
the  extremities  of  the  side  components  by  the  line  be,  cut- 
ting  7>Vi  in  /,    This  point  is  the  instantaneous  centre  about 
which  ns  is  rotating;  and  by  reason  of  the  connection  be- 
tween 7i7'"and  MS,  it  is  clear  that  EF  must  also  rotate  about 
/,  in  the  same  direction  and  with  the  same  angular  velocity* 
Therefore  the  motion  of  F  has  a  component  of  rotation  FK 
perpendicular  to  /•/,  the  angle  /7A~  being  made  equal  to  the 
angle  EIe\  and  also  a  component  of  translation  /•'//  parallel 
and  equal  to  Eh.    Completing  the  parallelogram,  the  motion 
of/f>is  the  diagonal  FG.     This  has  a  component  FJ  in  the 
line  of  77%  to  which  /X?  is  equal;  then  draw  at  (J  a  perpen- 
dicular to  PF  cutting  7YJ,  the  line  of  travel,  in  A',  and  7ltf  is 
the  resultant  motion  of  P. 

40.  Now,  the  point  E  is  common  to BS and  EF,  and  both 
pieces  have  the  same  instantaneous  centre  /;  consequently 


VELOCITY  DIAGRAMS.  4«> 

their  instantaneous  axes  must  lie  in  a  perpendicular  to  BS 
tli rough  /.  This  perpendicular  cuts  the  prolongation  of 
AB  in  G,  which  is  the  instantaneous  axis  of  £S\  and  it  cuts 
DE  in  0',  the  instantaneous  axis  of  EF.  The  actual  motion 
of  the  point  E  on  BS  is  EH%  the  resultant  of  the  components 
£ft  and  En'  equal  to  Bn,  and  OE  is  perpendicular  to  EH\ 
(incidentally,  the  sliding  at  E  is  equal  to  Eh  +  En') :  also, 
O'F*is  perpendicular  to  FG%  the  motion  of  F. 

Again,  the  actual  motion  in  space  of  the  line  EF,  is 
determined  by  the  two  motions  EL>  FG\  the  absolute  com- 
ponents of  EL>  as  above  seen,  are  Eh>  Ee> — those  of  FG  are 
Fg,  Fe\  Draw  gh  cutting  EF  in  /';  then  it  is  seen  that  the 
motion  ot  /•'/•' MUY  l>e  re^.inleil  as  a  rotation  about  /  as  an 
instantaneous  centre,  combined  with  a  longitudinal  transla- 
tion in  the  line  EF. 

41.  And  this  motion  is  equivalent  to  a  rotation  about  an 
instantaneous  axis,  which  must  lie  in  a  perpendicular  to  EF 
through   the  4>oint  /';   this  perpendicular  must,  therefore, 
pass  through   0\  which  has  already  been  shown  to  be  the 
instantaneous    axis.      Ami    since    all    the    lon^ it mlinal    eoui- 
ponents  must  be  equal,  it  follows  that  Fe'  =  EC.    Also  since 
at  /'  there  is  no  side  component,  the  absolute  motion  of  that 
point  is  in  the  direction  FE,  and  equal  to  Et*    In  the  small 
diagram  at  the  left,  this  motion,  7V",  is  seen  to  be   the 
resultant  of  a  rotative  component  /'£,  perpendicular  to  IF 
(the  angle  Flk  being  equal  to  the  angle  Efe),  and  a  com- 
ponent  of  translation  //A",  equal  and  parallel  to  Eh. 

42.  A  peculiar  device  for  producing  aggregate  motion, 
the  action  of  which  is  at  first  glance  rather  obscure,  is  shown 
in  Fig.  34.     It  consists  of  a  crank  AD>  turning  about  the 
fixed  centre  D  \  a  connecting-rod  AB,  and  a  lever  BC>  which 
vibrates  about  C  as  a  fixed  centre.    Upon  the  pin  B  are 


5° 


VELOCITY  DIAGRAMS. 


4* 


VELOCITY  DIAGRAMS.  51 

hung  the  wheels  Wand  X;  these  are  secured  together,  and 
turn  freely  on  the  pin,  and  X  engages  with  a  wheel  F,  which 
turns  freely  about  its  centre  C. 

A  wheel  U  is  secured  to  the  crank  AD,  and  communi- 
cates motion  to  W  through  the  intervention  of  an  idle  wheel 
V,  whose  bearing  is  the  pin  E,  fixed  in  the  connecting-rod. 

In  this  arrangement  the  crank  AD  is  the  driver,  and  the 
ultimate  follower  is  the  wheel  F;  given  the  velocity  of  the 
crank-pin  A,  then,  the  problem  is  to  find  the  circumferential 
velocity  of  that  wheel. 

It  needs  no  argument  to  show  that  if  this  problem  be 
attacked  analytically,  any  formula  expressing  the  action  of 
such  a  train  will  be  so  complicated  as  to  render  the  solution 
tedious,  if  not  difficult;  but  if  treated  graphically  it  is  both 
simple  and  easy,  and  requires  only  the  application  of  princi- 
ples already  explained. 

Supposing  the  crank  to  turn  to  the  right,  let  AM  repre- 
sent the  motion  of  A.  Resolve  AM  into  the  rectangular 
components  Aa,  Ab,  and  set  off  Ba'  equal  to  Aa  ;  draw  at  a' 
a  perpendicular  to  AB,  and  at  B  a  perpendicular  to  BC\ 
these  intersect  in  N,  and  BN  represents  the  motion  of  B. 
At  P,  the  point  of  contact  between  the  circumferences  of  X 
and  F,  draw  PQ  perpendicular  to  BC,  and  limited  by  its 
intersection  with  CN.  Suppose  for  the  moment  the  wheel 
W  to  be  removed  ;  there  would  then  be  nothing  to  cause  X 
to  rotate  about  B,  so  that  X,  F,  and  BC  would  move  as  one 
piece,  turning  about  C,  and  PQ  would  represent  the  circum- 
ferential velocity  of  the  wheel  F. 

But  when  W  is  restored,  this  value  PQ  will  be  affected 
by  two  things;  of  which  the  more  important  is  due  to  the 
rotation  of  F  about  its  centre  Et  caused  by  its  engagement 
with  U,  which  is  secured  to  the  crank  AD. 


52  VELOCITY  DIAGRAMS. 

In  order  to  determine  the  velocity  of  this  rotation,  draw 
TT,  tangent  to  U  and  V  at  their  common  point  F,  and  join 
bb'  the  extremities  of  the  side  components  of  AM  and  BN. 
Then  bb'  cuts  TT  in  (9,  and  FO  is  the  tangential  component 
of  the  point  F  on  the  circumference  of  V,  supposing  U'to  be 
removed.  But  Fis  also  a  point  on  the  circumference  of  U, 
and  its  absolute  motion  when  so  considered  is  FG,  perpen- 
dicular to  FD,  and  limited  by  constructing  the  angle  FDG, 
equal  to  the  angle  ADM.  Draw  677  perpendicular  to  TT\ 
then  FH  is  the  tangential  component,  and  OH  is  the  cir- 
cumferential velocity  of  F,  in  its  rotation  about  E — the  direc- 
tion being  as  indicated  by  the  arrow  v.  This  rotation  will 
impart  to  W  an  equal  circumferential  velocity  ;  therefore 
we  make  KJ,  the  common  tangent  of  Fand  W,  equal  to  OH. 
Draw  JB  to  limit  LS  tangent  to  X  and  parallel  to  JK,  then 
LS  should  be  in  this  case  added  to  PQ  on  account  of  the 
rotation  of  V  about  its  centre  E. 

43.  But  this  is  not  all ;  as  previously  stated,  the  value  PQ 
is  affected  by  another  thing,  which  is  due  to  the  motion  of 
the  link  AB  (carrying  V  with  it),  in  relation  to  BC.  This 
will  be  clear  when  it  is  considered  that  if  V  did  not  turn  at 
all  about  E,  but  moved  as  though  fixed  to  the  link,  the  point 
K  on  its  circumference  would  still  have  a  motion  Kf,  deter- 
mined by  the  intersection  of  bb'  with  KJ,  which  also  would 
be  imparted  to  W.  Draw./Z?  cutting  LS  in  s ;  it  will  then 
be  clear  that  PQ  must  is  this  case  be  increased  by  an  amount 
QR  equal  to  LS  -\-  Ls,  which  gives  PR  as  the  circumferential 
velocity  of  the  last  wheel  in  the  train. 

It  is  apparent  that  the  motion  Ls  is  due  to  the  folding  up 
of  the  link  AB  upon  the  lever  PC",  which  will  continue 
until  the  lever  reaches  the  position  €2,  the  extreme  limit  ot 
its  outward  excursion.  On  the  return,  the  directions  of 


VELOCITY  DIAGRAMS.  53 

both  PQ  and  Ls  will  be  reversed  ;  and  the  actual  velocity  of 
P  will  be  PQ  +  Ls  -  LS,  since  the  rotation  of  V  about  E 
goes  on  continuously  in  the  same  direction. 

It  is  also  evident  that  LS  is  always  greater  than  Ls ;  so 
that,  while  the  wheel  Y  turns  first  in  one  direction  and  then 
in  the  other,  it  is  driven  farther  to  the  left  than  to  the  right 
during  each  reciprocation  of  the  lever  BC,  and  consequently 
it  will  ultimately  make  complete  revolutions  about  its  cen- 
tre C,  in  the  direction  of  the  arrow  y. 

Now,  because  the  motion  of  a  point  in  the  circumference 
of  Y  is  reciprocating,  and  of  less  velocity  in  one  direction 
than  in  the  other,  it  follows  that  a  "  velocity  diagram  "  rep- 
resenting the  motion  will  be  of  the  general  iormAUCDE  in 
Fig.  35  ;  the  point  A  representing  the  instant  when  the  wheel 
begins  to  turn  in  the  direction  of  the  arrow,  the  point  C  the 
instant  of  reversal,  and  the  point  E  the  instant  when  it  again 
comes  to  rest.  Since  all  this  is  accomplished  during  one 
revolution  of  the  driving  crank,  AE  represents  the  time 
occupied  by  that  revolution,  of  which  the  velocity  is 
uniform. 

44.  Obviously  it  is  desirable  that  such  a  diagram  should 
begin  and  end  at  zero.  But  in  this  case  we  are  confronted 
at  the  outset  by  the  question,  at  what  position  of  the  crank 
is  the  wheel  at  rest  ?  Clearly  it  is  easy  enough  to  determine 
the  dead  centres  I,  2,  of  the  crank,  and  the  corresponding 
positions  I,  2,  of  the  pin  B ;  at  which  instants  the  lever  BC 
is  at  rest :  but  at  either  of  those  instants  the  wheel  Fwill  be 
found  to  have  a  definite  velocity,  which  may  be  determined 
by  the  processes  above  set  forth. 

In  such  a  case  the  construction  of  the  velocity  diagram 
is  the  readiest  means  of  determining  the  required  dead 
points  of  the  driven  wheel.  In  Fig.  36,  make  FL  equal  to 


54 


VELOCITY  DIAGRAMS. 


AE  of  Fig.  35  ;  place  the  crank  in  the  position  iZ>,  the  lever 
in  the  position  iC,  determine  the  velocity  of  P,  and  set  up 
the  ordinate  FG  equal  to  it;  then  the  ordinate  LM  will 
plainly  be  equal  to  FG.  Without  entering  into  the  details 
of  the  construction,  suppose  the  velocity  diagram  GHJM  to 

FIG.  35. 


H 


FIG.  36. 

be  drawn  ;  this  curve  will  cut  FL  at  the  points  7,  K,  which 
represent  the  instants  when  Y  is  at  rest. 

Then  supposing  the  crank-pin  A,  in  Fig.  34,  to  start  from 
point  i  upon  its  circular  path  (which  corresponds  to  F  in 
Fig.  36),  divide  the  circumference  of  that  path  into  parts 
proportional  to  Ff,  IK,  KL.  The  points  of  division  corre- 
sponding to  /  and  K  will  then  be  the  positions  of  the  crank- 
pin  at  the  instants  when  the  wheel  Y  is  at  rest ;  and  the 
positions  of  the  lever  BC  at  those  instants  can  then  be  read- 
ily found  in  the  usual  manner. 

Now  if,  in  Fig.  36,  we  set  back  FN  equal  to  LK  and  copy 


VELOCITY  DIAGRAMS.  55 

the  portion  MK  in  the  position  GN,  we  shall  have  the  curve 
NHIJK,  the  required  diagram,  identical  in  form  and  arrange- 
ment with  ABODE  in  the  preceding  figure. 

45.  A  further  example  of  planetary  wheel-work  forms 
part  of  the  combination  shown  in  Fig.  37. 

The  central,  or  "  sun,"  wheel  Wis  stationary,  as  shown 
by  the  screws  securing  it  to  the  frame. 

The  shaft  D  of  this  wheel  turns  freely  in  its  bearings, 
and  an  eccentric  is  keyed  upon  it,  in  which  are  the  bearings 
of  the  shaft  of  the  planet-wheel  W ,  whose  centre  is  6";  thus 
the  eccentric  itself  forms  the  train-arm.  The  shaft  C  has 
keyed  upon  it  the  wheel  W  at  the  back,  and  the  crank  CE 
is  in  front  of  the  eccentric  ;  the  eccentric-rod  is  pivoted  at£ 
to  a  sliding  socket  which  moves  upon  guides;  and  this 
socket  carries  a  pin  Ft  upon  which  turns  one  end  of  the 
lever  FP ;  and  this  lever  is  connected  with  the  crank  CE  by 
a  link  EH.  Nowr,  supposing  the  eccentric  to  turn  about  D 
at  a  given  rate,  it  is  required  to  determine  the  direction  and 
velocity  of  P's  motion  at  the  instant  when  the  parts  occupy 
the  position  shown. 

The  centre  of  the  eccentric  is  A  ;  let  AM  be  the  motion 
assigned  to  it;  then  CM', the  velocity  of  C,  is  found  as  shown 
by  prolonging  DM. 

Now,  the  crank  CE  and  the  wheel  W  being  virtually  one 
piece,  are  at  the  instant  rotating  about  the  instantaneous 
axis  /,  the  point  of  contact  between  the  sun-wheel  and  its 
planet. 

Therefore,  drawing  IM' ,  IE,  and  EL  perpendicular  to  IE, 
then  EL,  limited  by  making  the  angle  EIL  equal  to  the 
angle  C1M',  will  be  the  motion  of  E-,  this  has  a  component 
Eg  in  the  direction  of  EH,  to  which  Hg'  in  the  same  direc- 
tion must  be  equal.  Draw  AB  and  produce  it,  making  Ba* 


VELOCITY  DIAGRAMS 


VELOCITY  DIAGRAMS.  57 

equal  to  Aa,  the  component  of  AM  in  the  line  AB  ;  and  draw 
a'N  perpendicular  to  Ba'  to  determine  BN,  the  motion  of  B, 
to  which  the  motion  FJ  of  the  point  F  is  parallel  and  equal. 
Then,  with  reference  to  FP,  FJ  has  the  side  component  Fc, 
and  the  longitudinal  component  Fd\  and  both  Hdr  and  Pd" 
in  the  direction  FP  must  be  equal  to  Fd. 

Draw  at  d '  a  perpendicular  to  FP,  and  at  g1  a  perpendic- 
ular to  EH;  these  intersect  in  K,  and  HK  is  the  resultant 
motion  of  H,  which  has  a  side  component  He.  The  prolon- 
gation of  ce  will  limit  the  side  component  Pfoi  the  motion 
of  P;  the  longitudinal  component  is  Pd" ,  and  the  diagonal 
PR  of  the  completed  rectangle  is  the  required  resultant 
motion  ;  and  JKR  will  be  a  right  line  parallel  to  cef,  if  the 
construction  has  been  correctly  made. 

46.  In  Fig.  38,  W  is  the  pitch  circle  of  a  wheel  formed  in 
one  piece  with  an  eccentric  keyed  upon  the  shaft  D  ;  for  the 
sake  of  avoiding  confusion,  this  pitch  circle  is  drawn  of  the 
same  diameter  as  the  eccentric. 

As  in  the  preceding  figure,  the  eccentric  rod  is  pivoted 
at  B  to  a  piece  which  slides  upon  guides,  and  carries  a  pro- 
jecting pin  G,  upon  which  turns  the  lever  GP.  Then  A 
being  the  centre  of  the  eccentric,  and  A M  representing  its 
motion,  the  motions  £>N  and  its  equal  £Fare  determined  as 
in  Fig.  35,  so  that  no  more  need  be  said  of  them. 

The  eccentric  rod  in  this  case  has  a  projecting  portion 
on  the  upper  side,  in  which  is  fixed  a  pin  C,  with  a  wheel  w 
turning  freely  on  it ;  and  w  engages  with  W.  The  smaller 
wheel  has  also  a  projection  from  its  front  face,  in  which  is 
fixed  a  pin  E  by  which  a  link  is  pivoted  to  it,  whose  other 
end  is  pivoted  at  P  to  the  lever  GP.  Then,  knowing  AM, 
we  are  to  ascertain  the  motion  of  Pin  velocity  and  direc- 
tion. To  begin  with,  since  GVis  already  known,  we  find 


VELOCITY  DIAGRAMS, 


VELOCITY  DIAGRAMS.  59 

with  reference  to  GP  its  side  and  longitudinal  components, 
Gh  and  Gm ;  then,  since  Pm'  must  be  equal  to  Gm,  it  remains 
to  find  the  absolute  component  of  the  motion  of  P,  or,  what 
is  the  same  thing,  of  E,  in  the  line  of  the  link  EP. 

Now,  considering  the  wheel  w  by  itself,  it  turns  on  its 
bearing  at  C  solely  by  reason  of  its  engagement  with  W\ 
so  that,  if  the  latter  were  removed,  it  would  move  as  one 
piece  with  the  eccentric  rod.  And  the  motion  of  that  rod 
consists  in  a  rotation  about  /  (the  intersection  of  AB  with 
the  line  bb'  joining  the  extremities  of  the  side  components 
of  the  motions  of  A  and  B),  combined  with  a  translation  in 
the  direction  AB, — and  the  magnitude  Aa  of  the  latter  is 
already  known. 

The  wheels  Wand  w  touch  each  other  at  the  point  O  on 
AC',  then  Oc,  perpendicular  OI,  of  such  length  that  the 
angle  OIc  is  equal  to  the  angle  Alb,  is  the  rotative  compon- 
ent of  the  point  O  on  the  connecting-rod  ;  the  component  of 
translation  is  Oa"  equal  and  parallel  to  Aa,  and  OQ,  the 
diagonal  of  the  completed  parallelogram,  is  the  resultant 
motion,  which  would  be  the  same  for  the  point  O  on  the 
smaller  wheel  were  the  larger  one  removed. 

47.  But  the  motion  of  the  point  O  on  the  wheel  W,  is  OS 
perpendicular  to  OD,  limited  by  making  the  angle  ODS 
equal  to  the  angle  ADM.  Now  draw  TT,  the  common  tan- 
gent to  the  two  wheels,  and  upon  it  let  fall  the  perpendicu- 
lars QE' ,  SJ'\  these,  of  course,  should  be  equal  to  each  other 
and  to  the  normal  components  of  the  motions  OQ,  OS.  The 
tangential  components  OE'y  OJf,  lie  in  the  same  direction,  so 
that  their  difference  EJ'  represents  the  circumferential 
velocity  of  the  rotation  of  w  about  C.  For  the  sake  of  per- 
spicuity we  have  placed  the  point  E  upon  the  circumference 
of  w,  so  that  EJ,  perpendicular  to  CE,  and  equal  to  E'J' ,  is 


60  VELOCITY  DIAGRAMS. 

one  component  of  the  motion  of  £,  and  is  due  solely  to  the 
engagement  of  the  two  wheels.  But  were  W  removed,  E 
would  be  to  all  intents  and  purposes  a  point  of  the  connect- 
ing-rod ;  and  its  motion  might  be  determined  (since  the 
instantaneous  axis  is  inaccessible)  just  as  OQ  was,  which  is 
the  method  here  exhibited  ;  Ed  perpendicular  to  El,  making 
the  angle  Eld  equal  to  the  angle  Alb,  is  one  component,  while 
Ea'"  equal  and  parallel  to  Aa,  is  the  other  ;  then  EK  the 
diagonal  of  the  completed  parallelogram  is  the  resultant 
motion  upon  this  supposition.  Finally,  complete  the  paral- 
lelogram JEKL,  and  its  diagonal  EL  is  the  resultant  motion 
of  E  when  the  whole  mechanism  is  set  in  action ;  this  has  a 
component  El  in  the  line  PE,  to  which  PI'  must  be  equal ; 
erecting  at  I'  and  m'  perpendiculars  to  PE  and  PG,  respec- 
tively, they  intersect  in  R,  and  PR  is  the  resultant  required. 
In  reference  to  PG,  the  side  components  at  P  and  G  are 
Pg,  Gh ;  draw  gh  cutting  PG  in  f,  and  at  this  point  erect  a 
perpendicular  to  PG  ;  this  should  pass  through  F,  the  instan- 
taneous axis  of  PG,  which  is  the  intersection  of  PF,  GF, 
respectively  perpendicular  to  PR  and  GV. 


SIMULTANEOUS   DEAD-POINTS   IN   LINK-WORK. 

48.  In  the  train  of  link-work  shown  in  Fig.  39,  the  driv- 
ing crank  DA,  by  means  of  the  connecting-rod  AB,  imparts 
a  vibratory  movement  to  the  lever  CE\  which,  in  its  turn, 
causes  the  bent  lever  FGH  to  vibrate  through  the  interven- 
tion of  the  link  EF\  finally,  to  this  bent  lever  is  pivoted  the 
link  HP,  whose  extremity  P  is  constrained  by  guides  (not 
shown)  to  travel  in  the  vertical  line  IL.  And  the  problem 
is,  assigning  any  velocity  to  the  crank-pin  A,  to  determine 
the  resultant  velocitv  of  P. 


VELOCITY  DIAGRAMS. 


61 


Now  this  combination,  consisting  as  it  does  of  simple 
levers  and  links,  presents  no  new  feature,  except  in  the  two 
critical  positions  when  the  crank-pin  A  reaches  either  a 


FIG.  39. 

or  a'.  The  proportions  here  given  are  such  that  CG  = 
CE  —  EF-\-FG\  so  that  when  A  reaches  either  of  the 
above-mentioned  points,  the  centre  lines  CE,  EF,  and  FG, 
coincide  in  one  line  CfeG:  the  link  being  thus  folded  up 
upon  both  levers,  we  have  two  simultaneous  dead-points. 


> 


62 


VELOCITY  DIAGRAMS. 


It  is  stated  by  Prof.  Rankine  ("  Machinery  and  Mill- 
work,"  p.  193),  that  in  these  circumstances  the  ratio  of  the 
angular  velocities  of  the  two  levers  is  indeterminate ;  were 
this  so,  then  the  ratio  between  the  linear  velocities  of  the 
points  E  and  F  would  also  be  indeterminate.  I  propose  to 
show  that  neither  of  these  things  is  true. 

Since  the  motions  of  E  and  F,  in  the  position  under  con- 
sideration, are  both  perpendicular  to  the  line  joining  those 
points,  the  motion  of  the  link  must,  at  the  instant,  be  one  of 
rotation  about  some  point  on  EF  or  its  prolongation,  and 
the  first  step  is  to  find  that  point ;  the  method  of  doing  this 


FIG.  40. 

is  illustrated  in  Fig.  40.  The  upper  part  of  this  figure  gives 
a  side  view  of  two  levers  CA,  DB,  connected  by  a  link  AB, 
at  the  instant  of  collapse.  In  the  movement  diagram 
below,  it  must  be  understood  that  CV,  DY,  do  not  represent 
positions  of  the  levers,  but  that  AV,  BY,  and  HW,  represent 


VELOCITY  DIAGRAMS.  63 

the  motions  of  the  points  A  and  B,  and  of  H  the   required 
instantaneous  axis,  at  the  critical  instant. 

The  actual  magnitudes  of  these  lines  are  immaterial,  but 
the  relative  velocities  must  be  such,  and  H  must  be  so  situ- 
ated, in  all  cases,  as  to  satisfy  the  following-  conditions,  viz.: 

1.  Because  the  link  AB  is,  at  the  instant,  turning  about 
H  as  a  centre,  the  line    FF,  or  its  prolongation,  must  pass 
through  H. 

2.  Because  H  is  the  virtual  intersection  of  the  centre 
lines  of  the  levers  AC  and  BD,  the  magnitude  of  HW  must 
be  such  that  the  prolongations  of  CV  and  DY  shall  pass 
through  W. 

49.  The  lines  of  this  diagram  form  three  pairs  of  similar 
right-angled  triangles,  from  which  the  following  propor- 
tions are  readily  deduced,  viz.  : 


HW~fC  AV     _HA     _HD      AC 

HW      HD   f  "  BY'  ~~~  HB'  =~  HC  X  BD' 


BY"  BD 
Or,  substituting  the  symbols  given  in  the  figure, 

*          R  —  l—x      r_t 

J+~/-      x  +  r       <  R' 
whence 

3          2rl  Rlr  -  rr 

-~R^~r*'-       R  +  r   ' 
and  this  gives 


_  ±  VRlr(R  —  l+r)  —  rl 
R+r 

Which  proves  that  the  distance  of  the  instantaneous  axis 
H  from  the  point  A  is  not  indeterminate,  but  must  have  one 


64 


VELOCITY  DIAGRAMS. 


of  two  definite  and  easily  ascertained  values :  and  the  ratio 
of  the  linear  velocities  of  the  points  A  and  B  is  the  same  as 
the  ratio  of  their  distances  from  the  instantaneous  axis, — i.e., 


AV 
BY 


HB 


In  the  figure  the  proportions  are  as  follows :  R  =  6r 
r  =  3,  /  =  2.25,  and  the  resulting  values  are,  AH '=  -f  1.087, 
AH  =  —  2.587.  The  first  value  is  the  one  taken  in  Fig.  40  ; 
the  result  of  taking  the  second  is  shown  in  Fig.  41.  It  is 


apparent  that  the  diagrams  can  not  be  constructed  with  cer- 
tainty until  these  values  have  been  computed ;  though  the 
rudest  sketch  will  serve  the  purpose  in  making  the  calcula- 
tions. 

50.  The  ratio  of  the  linear  velocities  being,  then,  deter- 
minate, and  the  lengths  of  the  levers  fixed,  the  ratio  of  their 
angular  velocities  must  also  be  determinate.  In  order  to 
find  its  value,  let  v  represent  the  angular  velocity  of  R,  vr 
represent  the  angular  velocity  of  r\  then,  since  angular 

linear  velocitv 
velocity  = -p -,  we  shall  have 


_ 

'' 


VELOCITY  DIAGRAMS.  65 

IT'    I          v_ 

A    T 7"        I        *  *TI' 


_^ 
V  X  R  ~~  HA  X  R' 


Rx 

whence,  substituting  in  this  equation  the  value  of  *,  already 
found,  and  reducing,  we  have  finally 


±r  VRlr(R  — 


v'       ±R  VRlr(R-l-\-r)  -  Rlr 

This  expression,  with  the   assigned   lengths  of  the  link 
and  levers,  gives  the  values, 

v*  =  50  (very  nearly)>  for  Fi&-  40. 

.  41. 


51.  Now  a  question  of  at  least  theoretic  interest  arises 
in  regard  to  the  action  of  the  mechanism  shown  in  Fig.  39. 
It  is  usually  held,  I  believe,  that  only  the  component  in  the 
line  of  the  link  is  effective  in  transmitting  either  force  or 
motion  from  one  lever  to  another.  And,  so  long  as  there  is 
any  such  component,  there  is  no  occasion,  and,  it  may  be 
said,  no  ground,  for  questioning  the  correctness  of  that 
dictum.  Nor  yet  in  the  case  of  a  single  dead  centre  ;  thus, 
in  Fig  39,  the  link  AB  merely  rotates  about  its  instantaneous 
axis  B\  which  is  at  the  instant  stationary,  for  the  simple 
reason  that  it  can  not  move  both  ways  at  once  :  so,  although 
there  is  no  component  in  the  line  of  the  link,  no  motion  nor 
force  is  transmitted  to  the  lever  CBE. 


66  VELOCITY  DIAGRAMS. 

But  we  may  imagine  BD  in  Fig-.  40  to  drive,  with  a  con- 
stant  velocity,  for  a  limited  distance  on  each  side  of  the  crit- 
ical position  ;  in  that  case  the  velocity  of  AC  will  be  variable, 
of  course,  but  it  will  not  vanish  at  the  instant  of  collapse, 
notwithstanding  the  fact  that  there  will  then  be  no  compon- 
ent in  the  line  of  the  link — for  it  has  been  shown  that  it  must 
have  a  definite  value  as  compared  with  that  of  BD.  This 
arrangement,  then,  seems  to  present  a  remarkable  exception 
to  the  general  proposition  referred  to,  whether  regarded 
[dnematically  or  dynamically. 

52.  It  may  be  (and  has  been)  said  that  owing  to  the  elas- 
ticity of  materials,  to  the  impossibility  of  securing  absolute 
freedom  from  "  play  "  or  lost  motion, — in  short,  to  the  per- 
verse nature  of  inanimate  things  and  to  all  that  prevents  the 
realization  of  abstract  desiderata,  there  will  practically  be  a 
small  rotation  about  A  at  the  critical  instant ;  after  which 
there  would  be  a  longitudinal  component  of  motion  or  of 
force.     Were    this   the    true    explanation,    this    component 
would  be  very  small  in  comparison  with  its  effective  lever 
arm,  unless  the  rotation  about  A  were  very  considerable ; 
from  which  it  would  follow  that  the  levers  would  "  go  hard  " 
across  the  line  of  centres ;  and  that  the  more,  the  better  the 
fitting.     So   that,  were   the  fitting  absolutely  perfect,  the 
levers  could  only  cross  the   line   of  centres  by  the  aid  of 
momentum,  even  if  friction  were  entirely  eliminated  ;  and 
if  once  stopped  upon  that  line,  no  force  applied  to  either 
lever  could  move  the  other. 

53.  But  a  study  of  the  action  ol  a  model  made  for  the 
Stevens  Institute  of  Technology  leads  me  to  the  conclusion 
that  the  exact  contrary  is  the  case.     The  proportions  of  this 
model  are  the  same  as  those  of  the  figures  accompanying 
this  article  ;  and  the  levers  pass  the  critical  position  with  the 


VELOCITY  DIGRAMS.  67 

greatest  ease.  From  which  it  would  seem  that  the  link  at 
that  instant  acts  as  a  lever  whose  fulcrum  is  the  instanta- 
neous axis,  and  operates  by  side  pressure  instead  of  by  end- 
long thrust  or  pull. 

Another  interesting  point  in  regard  to  the  action  of  the 
model  is  this :  that  when  the  levers  are  placed  on  the  line 
of  centres,  the  mechanism  exhibits  no  hesitation  in  making 
a  choice  between  the  two  possible  positions  of  the  instanta- 
neous axis.  If  the  lever  BD  is  the  driver,  the  combination 
moves  as  indicated  by  the  diagram  in  Fig.  40  ;  but  when  AC 
is  the  driver,  the  motions  correspond  to  Fig.  41. 

It  may  be  added,  in  conclusion,  that  the  dead  points  may 
also  occur  simultaneously  when  the  two  levers  point  in  the 
same  direction ;  and  since  the  proportions  may  be  varied  in 
many  ways,  the  diagrams  may  present  appearances  very 
unlike  those  here  given.  But  this  method  of  reasoning  may 
be  applied  in  all  cases  ;  and  in  no  one  of  them  will  the  veloc- 
ity ratio  be  found  indeterminate. 

"  SLOW   ADVANCE   AND   QUICK   RETURN  "  MOTION  PRODUCED 
BY   ELLIPTICAL  WHEELS. 

54.  In  Fig.  42,  C  and  D  are  the  fixed  centres  of  a  pair  of 
elliptical  wheels,  in  contact  at  P.  In  order  that  these  may 
move  in  continuous  rolling  contact,  the  two  ellipses  must  be 
identical,  each  must  be  centred  upon  one  of  its  foci,  and  the 
distance  between  the  fixed  centres  must  be  equal  to  the 
major  axis  ;  in  practice,  both  wheels  must  be  provided  with 
teeth,  which  are  omitted  in  the  diagram,  since  the  action  is 
best  seen  and  discussed  by  consideration  of  the  pitch  curves 
only. 

Since  the  point  of  contact  must  always  lie  upon  the  line 
of  centres  CD,  its  motion  must  be  perpendicular  to  that  line, 


68  VELOCITY  DIAGRAMS. 

and  its  linear  velocity  is  the  same  whether  it  be  regarded  as 
belonging  to  one  ellipse  or  the  other,  so  that  the  angular 
velocities  are  inversely  proportional  to  the  contact  radii ; 
thus,  letting 

v  =  ang.  vel.  about  D,  and 
z/=    "       "         "        C,  we  have 

v       PC    , 

—f  —  7™  in  the  present  position, 

and  setting  off  the  equal  elliptical  arcs  PEK,  PIJ,  then,  when 
J  and  K  meet,  we  shall  have 

v      JC 
,  =  -j-r~  ;  or,  drawing  JC  and  joining  K  with 

F,  the  free  focus  of  the  right-hand  ellipse,  this  latter  value 
may  be  written 

v  _  KF 
*7  ~  KD' 

In  order  to  bring  K  and  J  into  coincidence,  it  is  apparent 
that  while  the  driver  turns  as  indicated  by  the  arrow, 
through  the  angle  PDK,  the  follower  must  turn  in  the 
opposite  direction  through  the  angle  PC/;  and  this,  it  is  to 
be  noted,  is  equal  to  the  angle  PFK. 

55.  These  ellipses  are  so  situated  and  proportioned  that 
PC  and  PF are  respectively  perpendicular  to  the  major  axes; 
and,  producing  these  lines  to  cut  the  ellipses  in  H and  G,  the 
arcs  Plffand  PEG  are  equal ;  whence  it  follows  that  while 
the  driver  turns  through  the  angle  GDP,  the  follower  will 
have  made  a  half  revolution. 

While  it  is  making  the  other  half,  the  driver  must  com- 
plete its  own,  thus  turning  through  an  angle  much  greater 


VELOCITY  DIAGRAMS.  69 

than  GDP\  since  the  two  ellipses  have  equal  perimeters  and 
necessarily  revolve  in  equal  times  about  their  fixed  centres 
•of  rotation. 

Now  let  a  pin,  fixed  in  the  follower  at  any  point  A  in  the 
major  axis,  actuate  a  link  whose  farther  extremity  B  is  con- 
strained by  guides  (not  shown)  to  travel  in  the  line  of  that 
axis,  then  the  traverse  of  B  along  that  line  will  be  equal  to 
AL  =  2AC  ;  it  will  make  the  upward  stroke  while  the  driver 
turns  through  the  angle  GDP,  and  the  downward  stroke 
while  the  driver  is  completing  its  revolution.  If,  then,  the 
driver  turns  uniformly,  the  down  stroke  of  B  will  occupy  a 
greater  time  than  the  up  stroke ;  which  is  just  what  is 
required  in  a  shaping  machine,  which  calls  for  a  "slow 
advance  "  of  the  tool-holder  while  making  the  cut,  and  a 
'"  quick  return  "  for  saving  time  while  no  work  is  done.  As 
here  arranged,  the  cut  would  be  made  on  the  down  stroke, 
and  by  the  "  pull "  of  the  link  ;  this  is  merely  for  the  sake 
of  saving  space  in  the  illustrative  diagram ;  and  it  will  be 
evident  that  had  B  been  below  A  instead  of  above,  the  con- 
dition would  have  been  reversed,  the  work  being  done  by 
the  "  push  "  of  the  link  ;  both  arrangements  have  been  used 
in  practice,  but  the  kinematic  action  is  the  same  whichever 
be  adopted. 

56.  Now  if,  in  the  design  of  a  shaping  machine  to  be  ac- 
tuated by  elliptical  wheels  as  above  shown,  the  ratio  of  the 
times  to  be  occupied  by  the  advance  and  the  return  be 
assigned,  the  first  thing  is  to  determine  the  eccentricity  of 
the  ellipses  which  will  satisfy  the  requirement :  the  manner 
of  doing  this  may  also  be  illustrated  by  Fig.  42.  Let  the 
time  of  the  return  be  to  the  time  of  the  advance  as  m  is  to  #, 
for  instance  ;  then  about  any  point  D  on  the  indefinite 
horizontal  line  HN  describe  a  circle  with  any  radius  MD. 


?o  VELOCITY  DIAGRAMS. 

Divide  the  upper  semicircumference  into  a  number  of  equal 
parts  represented  by  m  +  «,  and  set  off  MFequal  to  m  parts, 
then  FN '=  n  parts.  Draw  DF  and  produce  it ;  this  will  be 
the  direction  of  the  major  axis  of  the  driver :  draw  at  F  a 
perpendicular  to  DF,  cutting  HN  in  P;  then  P  is  a  point 
upon  the  ellipse,  of  which  D  and  F  are  the  foci,  and  the 
major  axis  is  equal  to  PF -\-  PD.  On  HN  set  off  PC  equal 
to  PF-,  produce  FP  to  A,  making  PA  equal  to  PD]  then  C 
and  A  will  be  the  foci  of  the  other  ellipse.  In  the  figure, 
the  advance  is  assumed  to  occupy  three  times  as  many  sec- 
onds as  the  return  ;  that  is  to  say,  m  =  i,  n  —  3,  consequently 
the  semicircumference  MFN  is  divided  into  4  equal  parts, 
and  MF  being  made  equal  to  one  of  these  parts,  FN  is  equal 
to  the  other  three. 

It  need  hardly  be  added  that  the  link  may  be  operated 
by  a  crank  keyed  in  any  convenient  position  upon  the  shaft 
of  the  driven  ellipse,  with  the  understanding  that  it  must  be 
upon  a  dead  centre  when  the  ellipses  occupy  the  relative 
positions  shown  in  Fig.  42. 

57.  Our  problem  now  is,  to  construct  a  diagram  repre- 
senting the  varying  velocity  of  B  throughout  the  revolution 
of  the  driving  ellipse,  whose  angular  velocity  is  assumed  to- 
be  constant.  In  order  to  do  this,  we  must  be  able  to  deter 
mine  at  any  given  instant  the  motion  of  the  crank-pin  A 
(which,  for  the  sake  of  simplicity  only,  has  been  placed  in 
the  focus  of  the  driven  ellipse).  The  direction  of  A's  motion 
is  known,  being  always  perpendicular  to  AC,  which  line 
revolves  about  C  in  a  direction  opposite  to  that  of  the  driver. 
In  determining  its  velocity,  we  have  only  to  remember  that 
angular  velocity  may  be  represented  by  the  linear  velocity 
of  a  point  at  unit  distance  from  the  axis  (the  magnitude  of 
this  unit,  for  purposes  of  comparison,  being  arbitrary) ;  and 


VELOCITY  DIAGRAMS.  71 

FIG.  43. 


FIG.  42. 


72  VELOCITY  DIAGRAMS. 

we  have  already  shown  that  the  angular  velocities  of  the 
driver  and  follower  are  inversely  proportional  to  the  radii 
of  contact,  at  any  given  instant. 

Assuming  then,  in  Fig.  42,  CA,  which  is  equal  to  DF,  as 
the  unit  distance,  and  assigning  to  F  any  linear  velocity  at 
pleasure ;  from  what  precedes  we  shall  have,  when  for 
instance  J  and  K  meet  upon  the  line  of  centres, 

KF-.KDr.lm.  vel.  Film.  vel.  A. 

The  required  velocity  of  A  may  then  be  easily  determined 
graphically  as  in  Fig.  43.  Upon  either  of  two  lines  inter- 
secting in  K  at  any  convenient  angle,  set  off  from  K  the  dis- 
tances KF,  ATZ^taken  from  Fig.  42 ;  on  the  other  line  set  off 
KV,  the  assigned  linear  velocity  of  F\  draw  VF,  and  through 
D  a  parallel  to  it,  cutting  the  prolongation  of  KV  in  V\ 
then  KV  is  the  required  linear  velocity  of  A,  since  by 
similar  triangles  we  have 

KF'.KD-.KV'.KV. 

58.  We  repeat  in  Fig.  44  the  construction  for  determin- 
ing the  velocity  of  B  when  that  of  A  is  known  ;  AC  repre- 
senting the  position  of  the  crank,  AB  that  of  the  link,  and 
CB  the  line  of  travel.  Let  AM,  perpendicular  to  AC,  repre- 
sent the  velocity  of  A  ;  this  has  a  component  Am  in  the  line 
of  the  link,  to  which  UN  must  be  equal.  Then  drawing  at 
N  a  perpendicular  to  AB,  cutting  CB  in  R,  we  have  BR  as 
the  resultant  velocity  of  B.  This  work  may  be  abbreviated 
by  drawing  AR'  parallel  to  CB,  and  limited  by  the  prolonga- 
tion of  Mm  perpendicular  to  AB,  since  the  triangle  AmR' 
thus  formed,  is  similar  and  equal  to  BNR. 

Now,  in  constructing  a  velocity  diagram  such  as  is 
required,  it  is  clearly  advisable  not  to  select  random  points 


VELOCITY  DIAGRAMS. 


73 


01  contact  between  the  driver  and  follower,  like  K  and  7  in 
the  preceding  illustrations,  but  to  proceed  in  an  orderly  and 


FIG.  45. 

systematic  manner.  For  the  purpose  of  showing  clearly 
how  this  may  be  done,  we  have  in  Fig.  45  reproduced  the 
driving  ellipse  in  the  same  position  as  in  Fig.  42 ;  PD  being 


74  VELOCITY  DIAGRAMS. 

the  initial  contact  radius,  and  PDG  the  angle  through  which 
the  driver  turns  during  the  "  quick  return  "  stroke  of  B. 
About  the  fixed  focus  D,  describe  a  "  measuring  circle  " 
large  enough  to  lie  well  outside  the  ellipse,  as  in  the  figure, 
and  produce  DP,  DG,  to  cut  its  circumference.  Divide  the 
quadrant  which  measures  the  angle  PDG  into  any  number 
of  equal  parts,  and  from  the  points  of  division,  I,  2,  3,  etc., 
draw  radii  to  D,  cutting  the  perimeter  of  the  ellipse  at 
a,  b,  c,  etc.;  from  these  points  draw  finally  the  lines  aF,  bF, 
etc.,  to  the  other  focus  of  the  ellipse. 

The  points  a,  b,  etc.,  will  evidently  become  contact  points 
at  equal  intervals  of  time  ;  and  as  each  one  does  so,  the  cor- 
responding positions  and  velocity  of  the  crank-pin  A  are  to 
be  determined  as  above  explained  in  reference  to  the  point 
JC  in  Figs.  42  and  43,  and  finally  the  velocity  of  B  for  each 
position  is  to  be  found  as  in  Fig.  44. 

59.  These  values  of  the  velocities  of  B  are  the  ordinates 
of  that  portion  XUY  vi  the  curve  in  Fig.  46,  which  pertains 
to  the  return  stroke.  Any  distance  XY  on  the  line  of  abscis- 
sas, representing  the  time  occupied  by  that  stroke,  is  divided 
into  as  many  equal  parts  as  the  quadrant  in  Fig.  45  ;  any 
point  of  subdivision,  as  b  for  instance,  represents  the  instant 
when  the  corresponding  point  of  the  ellipse  in  the  preceding 
figure  becomes  a  contact  point,  and  the  ordinate  represents 
the  velocity  of  B  at  that  instant. 

Since  the  advance  stroke  is  made  during  the  remaining 
three-fourths  of  the  revolution,  that  fraction  of  the  measur- 
ing circle  is  next  divided  into  any  convenient  number  of 
equal  parts  as  indicated  by  the  figures  i,  2,  3,  etc.,  inside  the 
circumference  in  Fig.  45  ;  in  Fig.  44,  YZ '==  ^XYt  is  similarly 
subdivided,  and  the  diagram  is  completed  bv  setting  up  at 
each  point  an  ordinate  representing  the  velocity  of  B  at  the 


VELOCITY  DIAGRAMS.  75 

instant  represented  by  that  point,  ascertained  by  construc- 
tions identical  with  those  already  explained. 

Now,  since  the  abscissas  represent  times  and  the  ordi- 
nates  represent  velocities,  and  since  also 

Time  X  Velocity  —  Space, 

it  is  clear  that  the  areas  of  the  two  curves  XUY,   YWZ, 
should  be  equal,  each  representing  the  length  of  the  stroke. 

60.  By  the  mode  of  construction,  as  seen  in  Fig.  42,  the 
major  axis  of  the  driving  ellipse  bisects  the  angles  through 
which  it  turns  during  both  strokes.     And  it  is  practically 
advantageous  to  subdivide  both  these  angles  so  that,  as  in 
Fig.  45,  the   successive    contact   points   upon  that   ellipse, 
which  are  made  use  of  in  determining  the  ordinates  of  the 
velocity  diagram,  shall  be  symmetrically  disposed  with  re- 
spect to  that  axis.     Because,  the  ratio  of  the  focal  distances, 
and  therefore  the  velocity  of  the  crank-pin  A,  will  be  the 
same  for  a  and  e,  for  b  and  d,  and  so  on,  thus  saving  much 
time  in  determining  the  velocities  of  A.     In  regard  to  the 
number  of  ordinates  to  be  determined,  it  is  impossible  to 
give   any  specific  directions ;   but,  in  constructing  these  or 
any  other  curves,  it  is  plain  that  the  points  through  which 
they    must  pass  should  be   nearest  each   other  where  the 
curvature  changes    most   rapidly;    as,  for   instance,  in  the 
region  of  U  the  vertex  of  the  return  stroke-curve  in  Fig.  46. 

61.  Another  diagram    of   interest   is   given   in    Fig.  47, 
where    the    curve    UVW  represents    the    varying   angular 
velocity  of  the  driven  ellipse,  the  constant  velocity  of  the 
driver  being  represented  by  the  horizontal  line  MN.    Since 
the  driver  and  the  follower  turn  in  opposite  directions,  it 
would,  perhaps,  be    more  consistent   to   place    UVW  and 
MN  on  opposite  sides  of  AB\  but  the  directional  relation  in 


76 


VELOCITY  DIAGRAMS. 


a  case  like  this  is  hardly  likely  to  be  lost  sight  of,  and  the 
varying  velocity  can  be  more  readily  compared  with  the 
constant  one  by  adopting  the  arrangement  shown  in  the 
figure. 

The  ordinates  of  this  curve  are  the  linear  velocities  of 


abode 


w 


<Y     1 


10     11 


FIG.  46, 


V 

\ 


N 


U 


c    d   e   G 
FIG.  47- 


n 


B,  as  determined  in  constructing  the  preceding  diagram. 
The  velocity  of  A  will  be  greatest  when  E  of  Fig.  42 
(corresponding  to  c  of  Fig.  45)  is  the  driving  point  of  con- 
tact, and  least  at  the  end  of  a  half  revolution,  when  the 


VELOCITY  DIAGRAMS.  77 

opposite  ends  of  the  major  axes  come  together.  In  the 
former  position  we  shall  have  for  the  velocity  ratio, 

-  =  -=rp,  and  in  the  latter  -  =  -^  ,  the  one  value  being  the 

reciprocal  of  the  other  ;  and  if  either  of  these  positions  be 
selected  as  the  initial  one,  it  is  clear  that  the  resulting  dia- 
gram will  be  symmetrical  with  respect  to  its  central  ordinate, 
because  the  ellipse  is  symmetrically  divided  by  its  major 
axis. 

We  prefer  to  make  the  first  ordinate  the  least,  as  AU\n 
Fig.  47,  where  AM  represents  the  constant  velocity  of  the 
free  focus  of  the  driver,  and  is  equal  to  KV  in  Fig.  43. 
Since  each  ellipse  completes  its  revolution  in  the  same  time, 
it  follows  that  the  area  of  the  curve  UVW  must  be  equal  to 
that  of  the  rectangle  AN,  and  that  the  areas  of  the  shaded 
portions  UMX,  XZc,  must  be  equal  to  each  other. 

62.  It  may  be  added  that  in  using  elliptical  wheels  for 
this  purpose,  the  writer  hereof  has  shown  that  any  given 
ratio  between  the  times  of  the  advance  and  the  return  may 
be  secured  by  means  of  ellipses  whose  eccentricity  varies 
between  certain  limits,  so  that  a  given  pair  of  elliptical 
wheels  might  be  used  for  shaping  machines  with  different 
conditions  as  to  the  relative  times  of  advance  and  return : 
in  the  course  of  the  investigation  it  was  also  demonstrated 
that  for  a  given  ratio  between  those  times,  the  construction 
illustrated  in  Fig.  42  gives  the  minimum  eccentricity. 

This  is  a  point  of  practical  importance  ;  for  since  it  is 
clearly  impossible  to  preserve  a  running  balance  in  the  ro- 
tating parts  of  mechanism  of  this  description,  it  is  as  clearly 
advantageous  to  reduce  the  unavoidable  vibration  to  the 
lowest  attainable  limit. 


78  VELOCITY  DIAGRAMS. 

OTHER  "  SLOW  ADVANCE  AND  QUICK  RETURN  "  MOTIONS. 

63.  In  Fig.  48  is  represented  the  device  known  as  "  the 
crank  and  slotted  arm,"  which  in  various  forms  has  been 
extensively  used  for  producing  the  slow  advance  and  quick 
return  motion  in  shaping  machines.  D  is  the  fixed  centre 
of  the  driving  crank,  whose  pin  P  turns  in  a  block  which 
slides  freely  in  the  slotted  arm  CA,  of  which  C  is  the  fixed 
centre.  To  the  extremity  A  of  this  arm  is  jointed  a  link, 
of  which  the  other  extremity  B  is  constrained  by  guides 
(not  shown)  to  travel  in  the  line  KH.  The  excursion  of  the 
arm  CA  is  plainly  limited  by  the  lines  CG,  CI,  tangent  at  E 
and  F  to  the  circular  path  of  P. 

With  G  and  /as  centres,  describe  with  radius  AB  arcs 
cutting  the  line  of  travel  of  B  in  //"and  K\  these  points  limit 
the  traverse  of  B  in  its  assigned  path.  When  P  is  in  the 
position  shown,  let  PO  perpendicular  to  PD  represent  its 
linear  velocity. 

Resolve  PO  into  components  PV  in  the  line  AC,  and  PL 
perpendicular  to  it;  the  former  represents  the  sliding,  which 
is  accommodated  by  the  slot,  and  the  latter,  PL,  is  the  linear 
velocity  of  the  point  Pin  the  line  CA,  in  rotation  about  C. 
Therefore,  AM,  perpendicular  to  AC,  and  limited  by  the 
prolongation  of  CL,  is  the  linear  velocity  of  A.  The  com- 
ponent of  AM  in  the  direction  BA  is  AN,  to  which  BQ  must 
be  equal,  and  a  perpendicular  to  BA  at  Q  cuts  the  line  KH 
in  R,  giving  BR  as  the  resultant  velocity  of  B. 

As  in  previous  cases,  this  may  be  abbreviated  by  drawing 

through  A  a  parallel  to  Kff,  cutting  MN  produced  in  T '; 

-.since  the  triangle  ANT  thus  formed  is  similar  and  equal  to 

BQR. 

»        64.  In  this  arrangement  it  is  obvious  that  the  arm  CA 


VELOCITY 

FIG.  48. 


79 


M     N        T 


O 


FIG.  480 


So  VELOCITY  DIAGRAMS. 

will  swing  through  the  arc  IG  while  the  driving  crank  turns 
through  the  arc  FPE,  and  A  will  return  from  G  to  /while 
the  crank  turns  through  the  remaining  part  of  the  circum- 
ference, EF. 

Consequently,  if  the  ratio  between  the  times  of  the 
advance  and  return  be  assigned,  as  for  instance  as  ;;/  is  to  n ; 
describe  a  circle  about  any  centre  D  with  any  radius  DP, 
divide  its  circumference  into  m  +  n  parts,  and  make  EF 
equal  to  n  parts;  then  EPF '  =  m  parts.  At  E  and  .Pthus  de- 
termined, draw  tangents  to  the  path  of  P,  intersecting  at  C ': 
this  determines  the  proportions  of  the  moving  pieces  and 
the  relative  positions  of  the  fixed  centres — it  being  obvious 
that  the  lengths  of  CA  and  of  AB,  as  well  as  the  direction 
of  fPs  travel,  are  entirely  arbitrary  in  theory,  although  in 
practice  they  are  limited  by  various  considerations  which 
need  not  here  be  discussed. 

The  link,  indeed,  may  be  dispensed  with  altogether  ;  thus 
in  Fig.  480,  the  outer  extremity  A  of  the  slotted  lever  is 
formed  into  a  pin  projecting  from  the  front  side  of  the  lever; 
this  pin  works  between  jaws  formed  or.  a  piece  S,  by  which 
the  cutting  tool  is  carried,  and  this  piece  is  constrained  by 
guides  (not  shown)  to  travel  in  the  direction  of  the  arrow. 
In  this  case,  the  motion  AM  of  the  point  A  is  to  be  resolved 
into  the  components  AX,  and  A  Y\  the  former  alone  is 
effective,  and  represents  the  resultant  motion  of  the  sliding 
head  S. 

65.  This  combination  has  also  been  used  in  a  curvilinear 
slotting  machine,  the  tool  being  carried  directly  by  a  curved 
extension  of  the  vibrating  arm,  as  shown  in  Fig.  49.  This 
figure  represents  an  application  to  the  work  of  reducing  the 
thickness  of  portions  of  the  rim  of  a  locomotive  driving- 
wheel  W,  when  the  latter  had  proved  not  t"»  be  in  correct 


VELOCITY  DIAGRAMS. 


81 


running  balance  :  as  described  and  illustrated  in  Professor 
Goodeve's  "  Elements  of  Mechanism." 

FIG.  49. 


FIG.  49<z. 


The  crank  is  here  replaced  by  a  disk,  provided  with  a 
slot  to  enable  the  pin  P  to  be  fixed  at  a  greater  or  less  dis- 
tance from  the  centre  D,  in  order  to  regulate  the  length  of 
the  stroke.  This  adjustment,  however,  affects  the  ratio 
between  the  times  of  the  advance  and  return  ;  as  the  effec- 
tive crank-arm  is  diminished,  the  relative  time  of  the  return 
will  be  increased.  This  will  be  clear  by  the  aid  of  Fig. 


82  VELOCITY  DIAGRAMS. 

where  it  is  obvious  that  the  angle  DCE'  is  less  than  the 
angle  DCE,  and  since  the  angles  at  E'  and  E  are  both  right 
angles,  it  follows  that  CDE'  is  greater  than  CDE. 

66.  In  Fig.  50  is  shown  the  modification  of  the  "  crank 
and  slotted  arm  "  device  known  as  the  Whitworth  motion ; 
it  differs  from  the  preceding  in  that  the  centre  C  lies  within 
the  path  of  P  instead  of  outside  of  it,  so  that  CA  makes  a 
complete  rotation  for  each  revolution  of  the  driving  crank. 
The  path  of  P  is  divided  at  E  and  F,  exactly  as  in  the  pre- 
ceding case,  for  the  purpose  of  securing  any  desired  ratio 
between  the  times  of  the  advance  and  return;   the  chord 
EF  is  then  bisected  in  order  to  locate  C ;  and  this  chord  is 
also  the  line  of  travel,  in  the  practical  use  of  the  movement. 
In  determining  the  speed  of  B  when  that  of  P  is  assigned, 
the  proceeding  is  precisely  the  same  as  in  Fig.  48,  and  since 
the   two   diagrams   are    lettered    similarly    throughout,    no 
further  explanation  is  required. 

67.  It  is  very  evident  that  the  device  shown  in  Fig.  50  is 
in  that  form  perfectly  impracticable  as  a  working  machine, 
since  the  shaft  C  would  interfere  with  the  crank  DP,  if  it 
projected   behind   CA,  while  if  it  projected  in  front  of  the 
slotted    arm  it  would  interfere  with  the  link  AB.     These 
difficulties    were   most   ingeniously    evaded    by  the    distin- 
guished  inventor,  and    an  extremely  serviceable    machine 
constructed  in  the  manner  illustrated  in  Fig.  51.    The  crank 
is  here  replaced  by  a  wheel  W,  riding  loose  upon  a  fixed 
shaft  whose  centre  is  D,  and  driven  by  a  pinion  w\  in  the 
front  face   of  this  wheel  is  fixed  the  driving   crank-pin  P. 
The  fixed  shaft  D  has  bored  in  it  a  hole  at  C,  as  shown  more 
clearly  in  Fig.  52.     Into  this  hole  projects  a  pin  forming  a 
part  of  the  piece  55,  Fig.  51,  at  the  back  of  which  is  a  slot 
in  which  slides  a  block  fitted  upon  the  pin  P.     In  the  front 


VELOCITY  DIAGRAMS. 


FIG.  51. 


FIG.  52. 


FIG.  50. 


84  VELOCITY  DIAGRAMS. 

face  of  55  there  is  also  a  slot  in  which  the  pin  which  drives 
the  link  AB  can  be  fixed  in  any  desired  position. 

By  varying  the  distance  between  A  and  C,  then,  the 
length  of  the  stroke  may  be  controlled  at  pleasure  ;  but  in 
this  case  there  is  no  change  in  the  ratio  between  the  times 
of  the  advance  and  the  return,  since  the  relative  positions 
of  P,  D,  and  C,  which  fix  this  ratio,  are  not  affected  by  any 
variation  of  AC. 

68.  Another  method  of  producing  a  slow  advance  and 
quick  return  movement,  by  the  use  of  levers  and  links  only, 
is  shown  in  Fig.  53.  In  this  combination,  the  driving  crank 
DA,  by  means  of  the  link  AB,  actuates  the  vibrating  lever 
BC.  To  the  extremity,  B,  of  this  lever  is  pivoted  the  link 
BP,  whose  free  end,  P,  is  constrained  to  move  to  and  fro  in 
the  path  HK. 

Supposing  as  before  that  the  time  of  traverse  in  one  di- 
rection along  HK\s>  to  be  to  the  time  occupied  in  the  return 
as  m  is  to  n,  the  skeleton  movement  is  laid  out  thus :  about 
any  centre  D  describe  a  circle  with  any  assumed  radius,  as 
DA  ;  and  divide  the  circumference  at  E  and  F  so  that 
EAF  =  m  parts,  while  FE  —  n  parts,  as  in  the  preceding 
cases.  Assuming  any  reasonable  length  for  the  first  link  AB, 
set  off  EDI  equal  to  this  assumed  length  ;  this  locates  the 
point  /,  the  crank  being  at  an  inward  dead  point.  Next 
draw  DF and  produce  it,  making/^ also  equal  toAB  ;  then 
G  is  the  extremity  of  this  link  when  the  crank  is  at  an 
butward  dead  point.  Draw  IG,  bisect  it  by  a  perpendic- 
ular, and  on  the  bisector  take  any  point  C  as  the  fixed  cen- 
tre of  the  vibrating  lever  CB\  in  practice  the  angle  ICG 
should  never  exceed  60°,  and  if  possible  it  should  be  less : 
the  length  of  BP  and  the  line  of  travel  of  P,  are  clearly 
arbitrary. 


VELOCITY  DIAGRAMS.  85 

This  being  merely  a  combination  of  links  and  levers, 
with  no  new  or  peculiar  features,  it  is  not  necessary  to  give 
a  detailed  explanation  of  the  process  of  determining  the 
velocity  of  P  when  that  of  A  is  given. 

69.  But  it  may  be  suggested  that  in  constructing  a 
velocity  diagram  for  either  of  these  combinations,  the  arc 
EF  should  be  subdivided  into  equal  parts,  and  the  remain- 
der of  the  circumference  then  divided  in  like  manner  by 


FIG.  53- 

itself;  it  might  at  first  glance  appear  more  expeditious  to 
divide  the  whole  circumference  at  once — but  the  proceed- 
ing suggested  will,  on  trial,  be  found  far  more  satisfactory. 

It  is  to  be  admitted  that  the  combination  in  Fig.  53  is 
not  particularly  suitable  for  use  in  a  shaping  machine;  but 
it  is  given  as  one  of  various  methods  of  producing  the 
required  motion, — and  moreover,  combinations  essentially 
the  same  in  principle  have  been  practically  employed  in 
various  forms  of  hot  air  and  gas  engines,  with  very  satis- 
factory results. 

70.  Thus  far  in  determining  the  motion  of  a  piece  driven 
by  a  lever  through  the  intervention  of  a  connecting-rod,  or 


86  VELOCITY  DIAGRAMS. 

link,  we  have  adhered  to  the  direct  process,  by  means  of  the 
longitudinal  component  of  the  motion  of  the  driving  point, 
because  the  reasoning  is  clear  and  simple,  and  perfect  famil- 
iarity with  that  method  is  essential.  We  now  proceed  to 
explain  one  or  two  other  methods,  which  in  some  cases  are 
more  expeditious,  although  the  reason  why  they  are  correct 
is  not  by  any  means  so  obvious. 

Fig.  54  shows  the  crank  and  connecting-rod  of  a  direct- 
acting  steam-engine,  the  cross-head  pin  O  travelling  in  the 
horizontal  line  OC.  At  C  erect  the  vertical  line  CX,  and 
produce  the  line  of  the  connecting-rod  OP,  if  necessary,  to 
cut  CX  in  the  point  A.  Then,  if  the  velocity  of  Pis  repre- 
sented by  a  line  equal  to  CP,  the  velocity  of  0  will  be  repre- 
sented by  a  line  equal  to  CA. 

For,  setting  off  PM  perpendicular  and  equal  to  PC,  and 
drawing  MB  perpendicular  to  PA,  and  PB  perpendicular  to 
CA,  the  triangle  PACis  similar  and  equal  to  the  triangle 
PMB.  And  it  has  already  been  shown  that  if  /W  is  the 
velocity  of  P,  PB  is  equal  to  the  velocity  of  O ;  and  CA  is 
equal  to  PB. 

If  for  any  reason  it  is  desirable  to  represent  the  velocity 
of  the  crank-pin  by  a  line  greater  or  less  than  CP,  as  for 
instance  CLt  it  will  now  be  obvious  that,  drawing  LK  par- 
allel to  OP,  we  shall  have  CK  equal  to  the  velocity  of  the 
cross-head  pin. 

71.  Fig.  55  also  represents  the  crank  and  connecting-rod 
of  a  direct-acting  engine.  In  this  construction,  PR,  the 
velocity  of  the  crank-pin,  of  any  magnitude  at  pleasure,  is 
set  off  on  the  prolongation  of  CP',  draw  through  R  a  par- 
allel to  PO,  cutting  at  T  n  vertical  line  through  O,  then  OT 
is  equal  to  the  velocity  of  O.  For,  first  setting  off  as  before 
PM  perpendicular  to  PC  and  equal  to  PR,  draw  PQ  parallel 


VELOCITY  DIAGRAMS. 


and  equal  to  OT,  MB  perpendicular  to  RQ,  and  PB  perpen- 
dicular to  PQ.  Then  the  triangles  PMB,  PRQ,  are  similar 
and  equal,  and  we  have  PQ  =  OT,=  PB. 

If  we  lay  off  the  velocity   CL  from  the  centre   C,  and 


FIG.  54- 


It 


FIG.  55- 


FIG.  56. 


draw  LK  parallel  to  PO,  cutting  the  vertical  through  C  in 
K,  the  triangles  CLK,  PRQ,  are  similar,  so  that  CK  will  be 
the  velocity  of  O,  as  in  Fig.  54. 


88  VELOCITY  DIAGRAMS. 

72.  In  Fig.  56  we  have  two  levers  DP,  CO,  connected  by 
a  link  PO.     The  construction  here  is  analogous  to  that  of 
the  preceding  figure ;  setting  off  PR,  the  assigned  velocity 
of  P,  on  the  prolongation  of  DP,  draw  through  R  a  parallel 
to  PO,  cutting  the  prolongation  of  CO  in  the  point  T:  then 
OT  is  the  velocity  of  O.     And  the  demonstration  is  also  the 
same  as  before ;  for  drawing  PM,  =  PR,  perpendicular   to 
DP,  draw    also   PQ   equal    and    parallel   to   OT,   MB   per- 
pendicular to  RQ,  and  PB  perpendicular  to  PQ ;  we   have 
then,  by  reason   of   the   two   similar  and   equal   triangles, 
PQ=OT,  =PB. 

In  order  to  save  the  reader  the  annoyance  of  referring 
to  preceding  diagrams,  we  have  in  Fig.  56  repeated  all  the 
steps  of  the  direct  determination;  thus,  PF is  the  absolute 
longitudinal  component  of  PM,  and  OE  is  equal  to  PF;  ON, 
the  resultant  motion  of  O,  is  perpendicular  to  CO,  and  is 
limited  by  drawing  EN  perpendicular  to  PO.  Then,  the 
triangles  OEN,  PFB,  being  similar  and  equal,  we  have 
PB  =  ON. 

And  thus  we  see  that  the  simplest  demonstration  that 
the  expeditive  processes  are  correct,  is  made  by  proving 
that  they  give  results  concordant  with  those  of  the  original 
direct  method. 

73.  We  have  already  explained  and  illustrated  the  rep- 
resentation  of  varying  angular  velocity,  in  the    case  of  a 
crank  rotating  continuously  in  the  same  direction.     That  is 
no  doubt  the  case  in  which  such  a  representation  is  most 
striking  and  most  readily  understood ;  but  the  same  meth- 
ods of  construction  are  equally  applicable  in  the  case  of  the 
circular  reciprocation   of   a  vibrating  lever,  although   the 
result  does  not  appeal  so  directly  to  the  eye,  and  the  dia- 
gram is  not  quite  so  easily  read. 


VELOCITY  DIAGRAMS.  89 

To  illustrate,  we  give  in  Fig.  57  a  skeleton  of  the  "  crank 
and  slotted  arm  "  movement,  like  that  in  Fig.  48  ;  and  in 
Fig.  58,  a  time-diagram  of  the  angular  velocity  of  the  vibrat- 
ing arm  AC.  The  slotted  arm  AC  moves  from  the  position 
CE  to  the  position  CF  while  the  pin  P  of  the  driving  crank 
travels  through  the  arc  EZF,  and  back  again  while  the  crank- 
pin  describes  the  remaining  arc  of  the  circumference,  FE\ 
in  this  instance  the  former  arc  is  twice  the  latter, — that  is,  if 
used  in  a  shaping  machine,  the  time  of  the  advance  will  be 
twice  that  of  the  return  :  consequently,  in  Fig.  58  we  make 


FIG.  58. 


HQ  —  2.QT.  In  Fig.  57  is  shown  the  process  of  determining 
one  ordinate  of  the  required  curve.  Since  angular  velocity 
=  linear  velocity  of  a  point  at  unit  distance  from  the  centre, 
the  first  step  is  to  describe  about  C  an  arc  BG  with  radius 
CB  —  DE.  In  this  particular  case  this  arc  will  be  tangent 
to  the  arc  EF,  because  EDC  —  60°,  whence  CD  =  2DE. 


90  VELOCITY  DIAGRAMS. 

74.  Now,  when  the   driving  crank  is  in  any  position   as 
DP,  assign  to  P  any  velocity  as  PO  ;  the  component  PL  per- 
pendicular  to  PC  is  the    velocity    of   the    point  P  on  PC. 
Then,  PC  cuts  the  arcHG  at  /;  and  IK,  perpendicular  to  PC 
and    limited    by  LC,   is  the  value   of  the  ordinate    sought. 
Then,  in  Fig.  58,  divide  HQ  so  that 

Hi  :  t'Q  : :  arc  EP :  arc  PZF, 

and  set  up  the  ordinate  ik  equal  to  IK. 

Now,  making  HM  =  PO  —  constant  velocity  P,  the  rec- 
tangle HN  represents  the  space  traversed  by  the  crank  in 
the  time  HQ,  which  is  the  arc  EZF\  and  the  area  of  the 
curve  HJQ  represents  the  space  traversed  by  the  point  B 
in  the  same  time,  or  the  arc  13G,  —  J  EZF\  therefore  the 
area  of  the  rectangle  should  be  four  times  that  of  the  curve. 
On  the  return  stroke,  the  ordinates  are  negative,  QR  is 
equal  to  PO,  and  since  GB  =  £  FE,  the  area  of  the  curve 
QVTis  half  that  of  the  rectangle  QS. 

It  is  hardly  necessary  to  point  out  that  the  curves  HJQ, 
QVT,  are  symmetrical  about  their  central  ordinates  JW, 
UV,  so  that  it  is  only  necessary  to  determine  the  ordinates 
for  one  half  of  each. 

75.  At  the  outset,  a  "  velocity  diagram  "  was  defined  as  a 
curve  whose  abscissas  represent  times,  while  the  ordinates 
represent    the  velocities,  linear   or   angular,   of   a   moving 
point  at  the  instants  indicated  by  the  feet  of  the  ordinates. 

This  is  what  is  distinctively  called  a  "velocity-time"  dia- 
gram, and  attention  has  purposely  been  thus  far  confined  to 
it  alone,  because  we  consider  it  to  be  the  most  explanatory, 
and  most  generally  useful,  method  of  graphically  exhibiting 
the  phenomena  of  varied  motion :  but  it  is  not  by  any 
means  the  only  method. 


VELOCITY  DIAGRAMS.  91 

In  Fig.  59,  we  have  shown  a  skeleton  of  the  "  Whit- 
worth  Motion  "  (Fig-.  50)  ;  DP  is  the  driving  crank,  CA  the 
slotted  arm,  AB  the  link,  BC  the  line  of  travel.  Assign  to 
P  any  velocity  at  pleasure,  determine  from  that  the  linear 
velocity  of  B,  and  set  down,  perpendicular  to  the  line  of 
travel,  the  ordinate  B21  equal  to  that  velocity.  Do  the 
same  for  as  many  different  positions  of  the  crank  and  arm 
as  may  be  deemed  necessary,  and  draw  a  curve  through  the 
extremities  of  the  ordinates,  for  both  the  advance  and  the 
return.  This  curve  is  technically  called  a  velocity-space  dia- 
gram, and  the  ordinates  of  course  show  the  velocity  of  B  in 
its  different  positions. 

These  same  ordinates,  it  is  clear,  are  the  ones  used  in 
constructing  the  "velocity-time"  diagram,  Fig.  60:  in  the 
former  the  abscissas  represent  varying  distances  traversed, 
in  the  latter  they  represent  the  equal  times  of  traversing 
them. 

76.  As  a  matter  of  interest,  we  give  in  Fig.  61  the  angu- 
lar velocity-time  diagram  for  the  same  movement ;  in  which 
AM  is  the  constant  angular  velocity  of  P,  and  the  curve 
UVW represents  the  varying  angular  velocity  of  A. 

It  is  obvious  from  inspection  6f  Fig.  59  that  the  mini- 
mum velocity  A  U  will  be  reached  when  both  the  crank  and 
the  slotted  arm  are  on  the  vertical  line  and  pointing  up- 
ward ;  and  the  maximum,  ZV,  after  a  half  revolution  from 
that  position,  when  both  point  downward.  The  ordinates 
of  the  curve  are  numbered  to  correspond  with  the  subdivi- 
sions of  the  circles  of  Fig.  59;  and  it  is  to  be  noted  that 
the  curve  crosses  the  horizontal  line  MN  at  the  point 
X,  coincident  with  the  ordinate  8-8,  at  which  instant 
the  slotted  arm  is  horizontal. 

The  ratio  of  the  time  of  advance  to  the  time  of  return 


92 


VELOCITY  DIAGRAMS. 


VELOCITY  DIAGRAMS. 


93 


is,  in  this  particular  case,  two  to  one 
to  observe  that  the  above  coincidence 
will  occur,  whatever  the  ratio.  This 
may  be  demonstrated  by  the  aid  of 
Fig1.  62,  where  PB  is  the  assigned 
velocity  of  P,  PF  the  component 
effective  in  driving  CA,  and  AR  the  re- 
sultant velocity  of  A.  We  have  then, 


but  it  is  of  interest 


PF  \PC\\PB\PD,  =  CA, 
.'.PF.  CA  =PC.PB\ 


also, 


PCiCA  ::PF:AR, 
.  •.  PC.  AR  =  PF.  CA,  =  PC.  PB,  FIG.  62. 

whence,  AR  =  PB Q.  E.  D. 

77.  Varying  velocity  may  also  be  represented  by  a  curve 
drawn  through  the  extremities  of  lines  of  different  lengths 
radiating  from  one  central  point,  forming  what  is  called  a 
polar  velocity  diagram.  Thus  in  Fig.  63,  the  circle  repre- 
sents the  path  of  the  pin  A  in  Fig.  59 ;  its  circumference  is 
similarly  subdivided,  and  on  the  radii  are  set  off  the  corre- 
sponding angular  velocities,  taken  from  Fig.  61.  The  re- 
sulting curve,  then,  serves  to  show  the  angular  velocity  of 
the  slotted  arm  in  any  given  position. 

In  Fig.  64,  the  circle  represents  also  the  path  of  A  in  Fig. 
59,  but  it  is  divided  into  equal  parts  like  the  path  of  P  in 
that  figure.  The  distances  set  off  on  the  radii  are  the  same 
as  in  Fig.  63,  and  the  polar  diagram  exhibits  the  angular 
velocities  of  the  slotted  arm  at  equal  intervals  of  time. 
And  it  is  quite  obvious  that  similar  diagrams  could  be  made 
to  represent  the  linear  velocities  of  A  at  different  points,  or 
at  different  times. 


94 


VELOCITY  DIAGRAMS. 


78.  Fig.  65  represents  the  crank,  link,  and  vibrating  lever 
movement  that  was  given  in  Fig.  53  as  one  means  of  pro- 


FIG.   65. 


ducing  a  slow  advance  and  quick  return  motion.   The  linear 
velocities  of  B  in  its  different  positions  are  laid  off  from  the 


VELOCITY  DIAGRAMS. 


95 


arc  IG  upon  lines  radiating  from  C,  so  that  the  resulting 
curve  is  in  effect  a  modified  velocity-space  diagram  in  which 
the  line  of  abscissas  is  a  circular  arc  instead  of  a  right  line. 

These  velocities,  as  shown,  are  determined  by  the 
method  of  Fig.  56 ;  and  this  figure  incidentally  shows 
another  demonstration  that  the  method  is  correct.  For, 
when  the  parts  are  in  the  positions  drawn  in  heavy  lines,  O 
is  the  instantaneous  axis  of  the  link  ;  consequently,  linear 
velocity  A  :  linear  velocity  B ' : :  OA  :  OB\  and  the  triangles 
OAB,  #3'3',  are  similar. 

79.  Still  another  mode  of  representing  varied  motion  is 
shown  in  Fig.  66,  the  movement  being  merely  the  crank, 
•connecting-rod,  and  cross-head,  of  a  direct-acting  engine. 


FIG.  66. 

The  circular  path  of  the  crank-pin  is  divided  into  equal 
parts  at  i,  2,  3,  etc.,  which  represent  equal  intervals  of  time; 
and  on  the  radii  through  these  points  are  set  out  from  the 
centre,  the  velocities  of  the  cross-head  pin  at  the  corre- 
sponding instants  :  and  the  curve  determined  by  these  points 
on  the  radii  is  for  some  occult  reason  designated  as  a  polar 
velocity-time  diagram  of  the  motion  of  the  cross-head  pin. 
The  linear  velocity  of  the  crank-pin  being  here  represented 


96  VELOCITY  DIAGRAMS. 

by  a  line  equal  in  length  to  the  crank,  the  required  veloc- 
ities are,  as  is  obvious  on  inspection,  determined  by  the 
method  of  Fig.  54. 

But  it  must  be  admitted  that  this  diagram  does  not  so 
instantly  appeal  to  the  eye,  or  explain  its  own  message  so 
clearly,  as  those  do  in  which  the  radial  ordinates,  as  in 
Figs.  63  and  64,  indicate  the  velocities  of  the  radii  them- 
selves in  circular  movement. 

80.  The  utility  of  velocity  diagrams   is    nowhere  more 
conspicuous  than  in  comparing  the  actions  of  different  com- 
binations which  effect  the  same  result. 

For  example,  Fig.  67  is  the  skeleton  of  an  oscillating 
engine,  with  the  trunnions  at  the  end  of  the  cylinder ;  in 
Fig.  68  the  trunnions  are  placed  at  the  middle  of  the  length 
of  the  cylinder  ;  and  Fig.  69  is  the  common  crank  and  con- 
necting-rod movement :  the  lengths  of  all  the  cranks  are 
equal.  Supposing  the  rotative  velocity  to  be  also  the' same, 
it  is  of  interest  to  compare  the  piston  speeds  with  each 
other,  and  also  with  the  piston  speed  of  another  engine 
having  an  infinite  connecting-rod,  or  slotted  cross-head,  of 
which  no  skeleton  is  given. 

The  velocity-time  diagrams  of  these  four  arrangements, 
for  one  stroke  from  left  to  right,  are  given  in  Fig.  70. 
That  for  the  engine  with  the  infinite  connecting-rod  is  the 
curve  marked  A  ;  this,  being  a  perfect  sinusoid,  is  symmet- 
rical about  its  central  ordinate  a,  which  is  also  the  maximum, 
and  equal  to  the  constant  velocity  of  the  crank-pin  ;  the 
crank-arm  being  at  that  instant  vertical. 

81.  Curve  No.  I  shows  the  piston  speed  of  the  oscillating 
engine  of  Fig.  67  ;  the  maximum  ordinate  c,  is  equal  to  a, 
but  since  this  corresponds  to  that  phase  of  the  movement 
when  the  piston  rod  is  tangent  to  the  path  of  the  crank-pin. 


VELOCITY  DIAGRAMS. 


97 


which  does  not  occur  until  after  the  crank  has  passed  the 

vertical  line,  this  ordinate  is  some  distance  to  the  right  of  a. 

The  engine  of  Fig.  68  gives  curve  No.  2  ;  this  is  quite 

similar  to  No.  i,  but  its  maximum  ordinate,  d,  is  still  further 


FIG.  69. 


to  the  right — the  reason  for  which  is  quite  apparent  from  a 
comparison  of  the  two  movements. 

The  engine  of  Fig.  69  gives  the  curve  No.  3,  which  is 
strikingly  different  from  either  of  the  others,  in  having  two 


98 


VELOCITY  DIAGRAMS. 


ordinates,  a  and  b,  which  are  equal  to  the  velocity  of  the 
crank-pin  :  the  maximum  ordinate  lies  somewhere  between 
these  two,  but  there  is  no  geometric  process  known  for  de- 
termining its  exact  location.  If,  however,  as  in  the  figure 
we  find  by  trial  and  error  the  centre  o  of  a  circle  which  sen- 
sibly agrees  with  the  curve  for  a  reasonable  distance  in  the 
region  of  the  vertex,  a  vertical  ordinate  k  through  the  cen- 
tre will,  it  is  safe  to  say,  be  as  near  to  the  precise  position 
as  is  necessary  for  any  practical  purposes.  Then  the  dis- 


FIG.  70. 

tance  between  a  and  k  will  be  a  certain  fraction  of  the  half- 
length  of  the  line  of  abscissas  ;  and  if  the  same  fraction  of 
the  quadrant  be  set  off  to  the  right  from  the  vertical  line  in 
Fig.  69,  the  phase  of  the  movement  at  which  the  piston  has 
the  maximum  velocity  will  be  practically  determined. 

But  the  phases  of  the  movement  at  which  the  piston 
speed  is  equal  to  the  crank  speed,  can  be  graphically  deter- 
mined in  a  very  simple  manner. 


VELOCITY  DM  GRAMS. 


99 


82.  Thus  in  Fig.  71,  the  crank  CA  is  vertical.  AB  is  the 
connecting  rod,  AM  the  velocity  of  A,**nd  Am  the  longi- 
tudinal component,  to  which  Bn  must  be  equal.  Drawing 
nN  perpendicular  to  AB,  it  is  evident  that  the  two  triangles 
AmM,  BnN,  are  similar  and  equal,  so  that  BN  =  AM.  And 
this  accounts  for  the  fact  that  in  Fig  70,  curve  No.  3  cuts 
curve  A  at  the  extremity  of  ordinate  a  ;  the  crank  being  ver- 
tical in  both  cases.  In  order  to  avoid  confusion,  the  other 
phase  is  shown  below  the  horizontal  centre  line  in  Fig.  71  ; 


FIG.  71. 

the  relative  positions  of  the  parts  being  such  that  the  pro- 
longation of  the  connecting-rod/*^  shall  pass  through  D,  the 
lowest  point  of  the  path  of  the  crank-pin.  Let  EG,  perpen- 
dicular to  CE,  be  the  velocity  of  the  crank-pin  ;  draw  GH 
perpendicular  to  FED,  also  EH  parallel  to  the  path  of  Fand 
consequently  perpendicular  to  CD,  then  EH  will  be  the  ve- 
locity of  F.  But  by  the  construction  the  triangles  CED,  GEH, 
are  similar  ;  and  since  CE  =  CD,  it  follows  that  HE=  EG. 

Now,  if  the  lengths  of  the  crank  CE  and  the  connecting- 
rod   EF  are  assigned,   there  is  no  geometrical  method  of 


loo  VELOCITY  DIAGRAMS. 

locating  E  or  F.  But  the  locations  of  both  points  may  be 
determined  with  great  accuracy,  by  marking  on  the  edge 
of  a  straight  slip  of  paper,  two  points  indicating  the  length 
of  EF.  Then,  moving  this  slip  around  so  that  the  point  E 
shall  always  lie  on  the  crank-path,  while  the  edge  of  the  slip 
passes  through  D,  mark  the  corresponding  location  of  F  for 
a  number  of  positions  of  E.  The  points  thus  marked  deter- 
mine a  curve  xy,  which  cuts  the  line  of  travel  in  the  point  F\ 
and  an  arc  about  this  point  as  centre,  with  radius  equal  to 
the  given  length  of  the  connecting-rod,  intersects  the  path 
of  the  crank-pin  at  the  required  point  E,  determining  the 
corresponding  position  CE  of  the  crank. 

Now,  the  differences  between  the  results  of  these  four 
arrangements  are  very  decided,  and  of  no  small  interest ;  and 
it  is  safe  to  say  that  these  differences  can  be  more  readily 
compared,  and  are  made  more  conspicuous,  by  the  dia- 
grams in  Fig.  70,  than  they  could  be  by  any  other  means 
whatever. 

83.  As  another  example,  let  us  consider  the  two  slow 
advance  and  quick  return  motions  shown  in  Figs.  72  and  73. 
The  first  is  the  Whitworth  movement,  which  has  already 
been  described ;  the  second  was  devised  by  the  writer,  with 
a  view  of  avoiding  the  sliding  in  the  slotted  arm  of  the 
former.  In  order  to  do  this,  the  two  cranks  are  connected 
by  a  short  link  AB  ;  and  the  method  of  construction  is 
shown  in  Fig.  74. 

Describe  about  D  a  circle  with  radius  DB  the  length  of 
the  driving  crank,  and  divide  its  circumference  so  that  the 
arc  BMF  shall  be  to  the  remainder  of  the  circumference  as 
the  time  of  the  return  is  to  the  time  of  the  advance ;  draw 
BF,  and  bisect  it  by  a  perpendicular,  which  locates  the 
fixed  centre  C.  Next  draw  the  link  BA  parallel  to  CD ; 


VELOCITY  DIAGRAMS. 


101 


102 


VELOCITY  DIAGRAMS. 


the  length  of  this  link  may  be  varied  within  narrow  limits, 
but  must  always  be  greater  than  MN  and  less  than  MO ; 
and  finally  draw  AC,  the  driven  crank.  If  now  we  suppose 
the  driver  to  turn  in  the  direction  of  the  arrow,  it  will  be 
seen  that  while  B  describes  the  arc  BMF,  A  will  have  been 
driven  through  the  arc  ANE,  or  180°,  the  two  cranks  then 
having  the  positions  DF,  CE ;  and  that  in  order  to  make  A 
describe  the  other  semicircumference  and  return  to  its 
original  position,  B  must  travel  through  the  remaining 
portion  FEB  of  its  own  path. 

This  combination,  clearly,  is  the  well-known  drag  link ; 


0       12345678 


FIG.  75. 

but  we  have  never  met  with  any  suggestion  of  using  it  for 
this  particular  purpose.  In  so  applying  it,  the  diameter 
ACE  must,  evidently,  be  the  line  of  travel  of  the  remote 
extremity  of  the  link  which  moves  the  tool-holder,  as 
shown  in  Fig.  73. 

Now,  these  two  movements  have  been  drawn  on  the 
same  scale,  and  constructed  to  have  the  same  proportion  be- 
tween the  times  of  advance  and  return  ;  and  there  is  a  pro- 
nounced difference  between  the  velocity-space  diagrams. 


VELOCITY  DIAGRAMS. 


103 


But  the  most  interesting  and  valuable  comparison  is  that 
between  the  velocity-time  diagrams,  which,  drawn  on  a 
larger  scale,  are  in  Fig.  75  shown  as  superposed ;  the  line 
of  abscissas,  and  the  scale  of  the  ordinates,  being  the  same 
for  each,  exactly  as  in  Fig.  70. 

84.  Again,  comparisons  between  angular  velocities  may 
be  advantageously  made  in  the  same  manner.  When  one 
rotating  piece  has  a  constant  angular  velocity,  this  fact  has 
already  been  illustrated  (see  Figs.  47,  58,  61).  But  in  Fig. 
76,  the  two  equal  reciprocating  levers,  BC  and  EF,  both 
move  with  varying  angular  velocities  ;  and  it  may  be  de- 
sirable to  compare  their  variations. 

The  driving  crank  AD  actuates  the  lever  BC  by  means 


E 


FIG.  76. 

of  the  link  AB,  thus  causing  B  to  traverse  the  arc  JHt 
exactly  as  in  Figs.  5 1  and  63  ;  and  BC  actuates  EF  by  means 
of  another  link  BE.  These  two  levers  were  made  of  the 
same  length,  it  may  be  stated,  simply  to  lessen  the  labor  of 
construction ;  for  taking  that  length  as  unity,  the  linear 
velocities  of  B  and  E  at  once  represent  their  angular  veloci- 


104 


VELOCITY  DIAGRAMS. 


ties.  And  the  values  of  these  velocities  are  the  ordinates 
of  the  two  curves  shown  in  Fig.  77,  the  one  drawn  in  the 
heavy  line  relating  to  the  lever  BC,  and  the  other  to  the 
lever  EF\  the  area  of  the  latter,  it  is  seen,  is  the  greater,  as 
it  obviously  should  be,  since  the  arc  KL  is  greater  than  HJ. 


FIG.  77. 

85.  Now,  if  we  divide  the  ordinates  of  one  curve  by 
those  of  the  other  in  their  order,  we  obtain  a  series  of  frac- 
tions ;  each  of  these  has  a  numerical  value,  which  is  either 
equal  to,  or  greater  or  less  than,  unity,  and  expresses  the 
value  of  the  velocity  ratio  at  the  corresponding  instant. 
And  by  setting  up  ordinates  with  these  values,  we  may 
construct  a  curve  exhibiting  the  variations  of  the  angular 
velocity  ratio.  By  inspection  of  Fig.  77,  it  will  be  seen  that 
the  value  of  this  ratio  will  be  unity  where  the  curves  in- 
tersect, as  at  6  on  the  right  and  at  j  on  the  left,  since  there 
the  ordinate  is  the  same  for  each. 

And  since  these  ordinates  become  more  nearly  equal  as 
we  approach  the  zero  points,  it  is  clear  that  unity  is  the 


VELOCITY  DIAGRAMS. 


I05 


limiting  value  when  the  ratio  becomes  — ,  as  it  does  when 

the  driving  crank  in  Fig.  76  is  on  the  dead  centres. 

In  the  nature  of  things  a  ratio  conveys  no  idea  of  direc- 
tion, 30  that  in  constructing  such  a  curve  the  ordinates  may 
all  be  set  up  on  the  same  side  of  the  line  of  abscissas,  thus 
requiring  but  one  line  for  unity  value;  and  this  has  been 
done  in  Fig.  78,  in  which  the  line  of  abscissas  is  equal  in 


ANGULAR  VELOCITY  RATIO  DIAGRAM 


87654321    08          7  6  5  4  3  2  0 

FIG.  78. 

length  to  that  in  Fig.  77,  and  subdivided  in  the  same 
manner. 

The  values  ot  the  ordinates  were  obtained  by  dividing 
those  of  the  curve  aaa  in  Fig.  77,  by  those  of  the  curve  bbb ; 
but  it  is  proper  to  state  that  all  this  work  has  been  done 
upon  a  small  scale,  and  for  illustrative  purposes  only,  and  of 
course  the  results  are  by  no  means  to  be  regarded  as  equal 
in  accuracy  to  those  exhibited  in  many  of  the  preceding 
diagrams 

This  then  shows  that  variable  velocity  ratios  may  be 
graphically  represented  by  a  curve.  But  a  ratio  is  essen- 
tially a  compound  idea;  and  though  its  value  at  any  instant 
may  be  represented  by  the  ordinate  of  a  curve,  as  in  Fig.  78, 
such  a  curve  conveys  no  idea  of  the  actual  velocities  which 


106  VELOCITY  DIAGRAMS. 

determine  that  value.  Of  the  two,  then,  Fig.  77  is  far  more 
explanatory  and  withat  more  readily  understood,  since  both 
terms  of  the  fraction  are  shown  at  the  same  time  and  on  the 
same  ordinate.  Still,  there  may  be  cases  in  which  a  com- 
parison of  variations  in  angular  velocity  ratio  might  be 
desirable  ;  and  then  the  construction  shown  in  Fig.  78  may 
prove  to  be  of  great  utility. 

ACCELERATION. 

86.  We  pass  now  to  the  consideration  of  some  graphical 
operations  which,  while  not  directly  connected  with  the 
construction  of  velocity  diagrams  as  heretofore  considered, 
are  nevertheless  closely  related  to  it,  and  are  of  at  least  equal 
interest. 

In  Fig.  79,  mark  on  the  line  of  abscissas,  beginning  at 
zero  on  the  left,  the  equidistant  points  i,  2,  j>,  etc.,  each  rep- 
resenting an  instant;  then  the  spaces  between  them  will 
represent  equal  periods  of  time.  Set  up  at  i  an  ordinate  la, 
representing  on  any  convenient  scale  the  distance  traversed 
by  a  moving  point  during  the  first  period.  Let  the  distance 
traversed  during  the  first  two  periods  be  represented  by  an 
ordinate  2b,  twice  as  great  as  la.  At  the  end  of  three 
periods,  let  the  distance  be  three  times  as  great,  at  the  end 
of  four,  four  times  as  great,  as  indicated  by  the  ordinates 
jc,  4dr  and  so  on.  It  will  then  be  obvious  that  the  line 
drawn  through  the  points  a,  b,  c,  d,  etc.,  will  be  a  right  line 
passing  through  the  zero  point;  and  also,  that  the  difference 
between  any  two  ordinates  will  represent  the  distance  trav- 
ersed by  the  moving  point  during  the  intervening  period 
—thus,  4d — la  =  hd,  the  distance  traversed  during  three  of 
the  equal  periods  first  set  oft.  Similarly,  drawing  through 


VELOCITY  DIAGRAMS. 


107 


c  the  horizontal  line  ce,  we  have  de  as  the  space  traversed  in 

the  time  ce,  and  since    -2 —  velocity,  —  represents  the 

time  J     " 


ce 


velocity  at  the  instant  j  ;  and  if  we  regard  ce  as  a  unit  of 
time,  de  itself  will  be  equal  to  the  velocity  ;  and  accordingly 
3d'  is  set  up  equal  to  dc.  In  this  case  the  velocity  is  clearly 
uniform — and  may  accordingly  be  represented  by  a  hori- 
zontal line  through  d! ';  as  it  is  quite  obvious  that  for  any 


FIG.  79. 

other  instant,  as  for  instance  /,  the  same  process  will  give 

for  the  velocitv,  —  =  — . 
ag        ce 

The  line  drawn  through  the  extremities  of  the  ordinates 
(in  this  case  ad)  may  properly  be  called  the  line  of  displace- 
ment;  and  it  is  straight  because  the  differences  between  suc- 
cessive equidistant  ordinates  are  equal.  Had  they  been  un- 
equal, that  line  would  have  been  a  curve;  as  for  example  xy ; 
and  let  us  suppose  the  conditions  to  be  such  that  xy  is  tan- 
gent to  ad  at  the  point  c.  The  velocity  of  the  moving  point 
at  the  instant  3,  being  measured  by  the  distance  which  it 
would  travel  in  a  unit  of  time  if  that  velocity  were  uniform. 


io8  VELOCITY  DIAGRAMS. 

would  clearly  be  the  same  as  that  just  determined.  And 
had  that  curve  of  displacement  been  assigned,  the  velocity 
at  that  instant  would  be  ascertained  by  erecting  an  ordinate 
cutting  the  curve  at  c,  drawing  through  that  point  a  tangent 
line  and  also  a  horizontal  line  of  a  length  ce  representing  the 
unit  of  time,  and  erecting  the  vertical  line  ed. 

87.  To  illustrate  a  practical  application  of  this  process, 
let  us  take  the  crank  and  connecting-rod  movement  of  Fig. 
80;  supposing  the  rotation  to  be  uniform,  the  circumference 
is  divided  into  equal  parts,  and  the  corresponding  positions 
of  the  cross-head  pin  are  determined  as  usual.  Then  the 
distances  01,  02,  03,  etc.,  are  set  up  as  ordinates  at  the  equi- 
distant points  i,  2,  j,  etc.,  in  Fig.  81,  the  spaces  between 
them  representing  equal  times;  and  thus  we  construct  the 
time  displacement,  curve  as  shown.  The  velocity  at  4  is 
determined  as  above  explained,  and  the  same  process  being 
repeated  for  the  other  instants,  we  have  a  velocity  diagram, 
derived  from  the  curve  of  displacement ;  the  whole  process 
being  entirely  different  from  anything  previously  described. 
And  setting  up  the  ordinates  of  this  curve  at  the  corre- 
spondingly numbered  positions  in  Fig.  80,  we  can  construct 
the  velocity-space  diagram. 

Now  in  order  to  test  the  accuracy  of  this  new  method 
by  comparing  it  with  the  old,  we  must  know  the  circumfer- 
ential velocity  of  the  crank-pin  ;  this  is  found  by  means  of 
the  fact  that,  as  has  .previously  been  shown,  it  is  equal  to 
that  of  the  cross-head  pin  when  the  crank  is  vertical,  that  is 
to  say,  to  the  ordinate  j-j.  Applying  this  test,  the  results 
of  the  two  methods  were  found  to  be  identical,  in  this  case  ; 
but  so  happy  a  coincidence  is  not  always  to  be  looked  for, 
since  the  new  process  depends  upon  the  accurate  drawing 
of  a  series  of  tangents,  which  in  dealing  with  a  curve  of 


VELOCITY  DIAGRAMS. 


109 


no  VELOCITY  DIAGRAMS. 

unknown  geometrical  properties  is  sometimes  quite  a  diffi- 
cult, not  to  say  an  uncertain  task;  particularly  if  the  curve 
is  very  flat. 

88.  As  to  the  theoretical  correctness  of  this  process, 
however,  the  argument  given  in  connection  with  Fig.  79 
would  appear  conclusive  ;  but  corroborative  evidence  is 
found  by  its  application  to  a  displacement  curve  of  known 
properties,  to  which  the  tangent  can  be  drawn  by  geometri- 
cal construction  ;  of  which  Fig.  82  is  a  good  example.  In 
order  to  give  the  diagram  a  more  convenient  form,  the 
times  are  set  up  on  the  vertical  line,  as  at  i,  2,  3,  etc.,  and 
the  corresponding  spaces,  as  i,  4,  9,  etc.,  are  laid  off  on  the 
horizontal  line,  and  the  time-unit  is  four  times  the  space- 
unit.  Since  the  spaces  are  proportional  to  the  squares  of 
the  times,  the  displacement  curve  is  a  parabola,  of  which  F 
is  the  vertex  and  the  horizontal  line  through  F  is  the  axis  ; 
the  tangent  to  which  at  any  point  is  most  conveniently 
drawn  by  aid  of  the  property  that  the  subtangent  is  bisected 
at  the  vertex.  To  draw  the  tangent  at  A,  then,  set  off 
FG  =  Fi6,  and  draw  GA.  By  the  same  process  as  in 

BC 

Fig.  8 1,  then,  we  have    — -  as  the  value  of  the  velocity  at 

B A 

the  instant  4,  and  BA  being  unity,  we  set  off  40,  —  BC. 

Now   to  find  the  numerical  value  of  BC,   we    have  by 

BC      FG        BC       1 6 
similar   triangles,  -5-:  =  -~~,  or =  -^-,  whence  BC  =  8. 

z>yi          r rL  4  ° 

If  this  process  be  repeated,  we  shall  find  in  a  similar  man- 
ner that  the  velocities  at  the  instants  i,  2,  j,  etc.,  are  re- 
spectively 2,  4,  6,  etc.,  showing  that  the  velocity  diagram 
is  a  straight  line,  and  agreeing  with  the  well  known  law  01 
falling  bodies,  that  the  velocity  varies  directly  with  the 
time,  and  the  space  with  the  square  of  the  time. 


VELOCITY  DIAGRAMS.  in 

89.  Now,  just  as  the  difference  between  two  consecutive 
ordinates  of  the  displacement  curve  indicates  the  velocity 
(.or  the  rate  at  which  the  displacement  is  changing),  so  the 
difference  between  two  consecutive  ordinates  of  the  velocity 
curve   indicates    the    acceleration,    or    rate    at    which    the 
velocity  is  changing.     And  by  applying  to  the  latter  curve 
the  same  process  that  was  applied   to  the  former,  we  may 
determine  an  acceleration  curve  as  in  Fig.   81,  exhibiting 
this  rate  of  change  at  each  instant  of  the  motion.     Thus  KP 
is  tangent  to  the  velocity  curve  at  K,  through  which  point 

PL 

the  horizontal  line  KL  is  drawn,  and  -777  represents  the  ac- 

J\.  ./_> 

celeration  at  the  instant  5;  since  KL  =  unit  time,  the  ordi- 
nate  5-5  is  made  equal  to  LP.  But  it  is  measured  down- 
ward from  the  line  of  abscissas,  because  P  is  below  the 
horizontal  KL,  indicating  that  the  velocity  is  undergoing 
retardation;  which  it  obviously  must  be,  since  the  maxi- 
mum velocity  was  reached  between  the  instants  3  and  4. 

In  Fig.  82,  the  time-velocity  is  represented  by  a  right 
line,  and  as  in  Fig.  79,  the  application  of  this  process  will 
produce  another  right  line,  parallel  to  the  axis  on  which 
the  times  are  measured.  The  distance  of  this  line  of  ac- 
celeration may  be  found  thus  :  By  similar  triangles  we  have 

be          DC          be        10 

— r  —  TVCM  or  —  =  — ,  whence  be  =  2  ;  which  accords  with 
ao  JJr  4  20 

the  other  law  of  falling  bodies,  that  the  acceleration  is  con- 
stant, and  equal  to  the  velocity  acquired  at  the  end  of  the 
first  instant,  counting  from  the  state  of  rest  as  zero. 

90.  This  process  may  be  and  has  been  described  as  one 
of  graphical  differentiation;  and  correctly,  too  ;  for  the  ratio 

MN 

-,  in  Fig-  81,  will  remain  the  same,  no  matter  how  short 


112  VELOCITY  DIAGRAMS. 

AM  may  be,  and  Avhen  A  and  M  become  consecutive,  the 

MN 
fraction    .  ...  will  become,  in  the  language  of  the  calculus, 

dy 

merely   -7-  ,  or  the  first  differential  coefficient  of  the  curve 

of  displacement,  from  which  the  velocity  curve  is  derived 
by  differentiation. 

PL 

And    similarly,       j  ,    representing   the  first   differential 


coefficient  of  the  second  curve,  represents  also  the  second 
differential  coefficient  of  the  original  one.  But  the  explana- 
tion given  in  connection  with  Fig.  79  will,  we  think,  make 
the  matter  clear  to  those  not  familiar  with  the  language  of 
analysis. 

It  is  to  be  observed  that  in  Fig.  81,  the  final  ordinate  66 
of  the  acceleration  curve  is  equal  to  TR,  to  obtain  which  a 
tangent  PT  \s  drawn  to  the  curve  at  its  extremity,  and  6R  is 
set  off  equal  to  the  unit  of  time.  Also,  PT  makes  a  greater 
angle  with  the  horizontal,  than  any  other  tangent  which 
slopes  downward  toward  the  right  ;  and  of  those  which 
slope  downward  toward  the  left,  the  greatest  angle  is  made 
by  the  tangent  at  the  zero  point.  Consequently,  the  maxi- 
mum ordinates  of  acceleration  and  retardation  respectively, 
correspond  to  the  beginning  and  the  end  of  the  stroke  ; 
which  is  clearly  as  it  should  be  in  this  case. 

91.  But  it  is  by  no  means  true  in  all  cases  ;  a  good  illus- 
tration of  this  is  given  in  Fig.  83,  where  the  velocity  curve 
is  that  of  the  "quick  return"  of  a  Whitworth  movement. 
Here  it  is  evident  that  the  inclination  of  the  curve  to  the 
horizontal  (which  is  measured  by  that  of  its  tangent),  in- 
creases as  we  recede  from  the  zero  point,  until  the  point  of 
contrary  flexure  is  reached,  when  it  reaches  its  maximum, 


VELOCITY  DIAGRAMS. 


1*3 


and  so  does  the  ordinate  of  the  curve  of  acceleration. 
When  the  velocity  is  at  a  maximum,  it  is  clear  that  the 
acceleration  will  be  nil ;  the  tangent  is  at  that  instant  hori- 


FIG.  83. 

zontal,  and  after  that  it  will  slope  downward  to  the  right : 
and  similar  reasoning  to  the  above  applies  to  the  negative 
acceleration,  or  retardation. 

Since  the  motion  begins  at  zero  and  ends  at  zero,  it  is 
clear  that  the  retardation  and  the  acceleration  must  exactly 
balance,  so  that  the  area  of  that  part  of  the  curve  above  the 
line  of  abscissas  should  be  equal  to  the  area  of  the  part 
below. 


INDEX. 


PAGE 

Absolute  Components 5 

Acceleration 106-113 

Axis,  Instantaneous 10 

Centre,  , n 

Components,  Absolute 5 

Longitudinal 4 

Normal 12 

Side 4 

Tangential 12 

Composition  of  Rectilinear  Motions 3 

Revolution  and  Rotation 26,  27 

Contact  Motions 12 

Rolling 34-39 

Dead  Points,  Simultaneous 60-66 

Differentiation,  Graphical m,  112 

Displacement,  Line  of 107 

Instantaneous  Axis 10 

Centre II 

Intersection  of  Two  Rotating  Right  Lines 18-22 

Curved   Lines 23 

Line  of  Displacement 107 

"  Pilgrim-step  "  Motion 50-54 

Planetary  Wheelwork 40,  42,  47 

Roberval's  Method  of  Drawing  Tangents 14 

Rolling  Contact 34~39 

Simultaneous  Dead  Points 60-66 


n6  INDEX. 

PAGB 

•*  Slow  Advance  and  Quick  Return  "  Motions 67-85 

By  Crank  and  Slotted  Arm 78-81 

Drag  Crank 100-102 

Elliptical  Wheels 67-77 

Links  and  Levers 84,  85 

Whitworth's  Motion 82,  83 

Velocity  Diagrams — Angular 75 

Velocity  Ratio 104,  105 

Comparison  of 97~i  4 

Polar 93,95 

Space 90,  91 

Time 2,  90,  91 


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mi.  I     MiiiliiiiiKiiK  Svo,  I  00 

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luiiiion.i  :;p.i/.i  ir.nio.  moroeob,    l  W 

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(///    />;•/•/»»/  MI  /Jon.) 

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Vol.  I.-Silver 8vo,  7  50 

Vol.  II.— Gold  and  Mercury 8vo,  7  50 

14 


Keep's  Cast  Iron ^flLATHCAS ^vo>  ^  ^ 

Kunhardt's  Practice  of  Ore  Dressing  in  Lurope 8vo,  1  50 

Le  Chatelier's  High- temperature  Measurements.     (Boudouard — 

Burgess.) 12mo,  3  00 

Metcalf s  Steel.    A  Manual  for  Steel-users 12mo,  2  00 

Thurston's  Materials  of  Engineering.    In  Three  Parts 8vo,  8  00 

Part  II.— Iron  and  Steel 8vo,  3  50 

Part  III.— A  Treatise  on  Brasses,  Bronzes  and  Other  Alloys 

and  Their  Constituents 8vo,  2  50 

MINERALOGY. 

Barringer's    Description    of    Minerals    of    Commercial    Value. 

Oblong,  morocco,  2  50 

Boyd's  Resources   of   Southwest  Virginia 8vo,  300 

"       Map  of  Southwest  Virginia Pocket-book  form,  2  00 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.)  .8vo,  4  00 

Chester's  Catalogue  of  Minerals 8vo,  paper,  1  00 

Cloth,  1  25 

Dictionary  of  the  Names  of  Minerals 8vo,  3  50 

Dana's  System  of  Mineralogy. ...   Large  8vo,  half  leather,  12  50 

"      First  Appendix  to  Dana's  New  "  System  of  Mineralogy." 

Large  8vo,  1  00 

"      Text-book  of  Mineralogy 8vo,  4  00 

Minerals  and  How  to  Study  Them 12mo,  1  50 

Catalogue  of  American  Localities  of  Minerals. Large  8vo,  1  00 

"      Manual  of  Mineralogy  and  Petrography 12mo,  2  00 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Hussak's     The     Determination     of     Rock-forming     Minerals. 

(Smith.)    Small  8vo,  2  00 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of 

Mineral  Tests 8vo,  paper,  50 

Rosenbusch's  Microscopical  Physiography  of  the  Rock-making 

Minerals.      (Idding's.) 8vo,  5  00 

*Tillman's  Text-book  of  Important  Minerals  and  Rocks.. 8vo,  2  00 

Williams's  Manual  of  Lithology 8vo,  3  00 

. 

MINING. 

Beard's  Ventilation  of  Mines 12mo,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  00 

Map  of  Southwest  Virginia Pocket-book  form,  2  00 

•Drinker's     Tunneling,     Explosive     Compounds,     and     Rock 

Drills 4to,  half  morocco,  25  00 

Eissler's  Modern  High  Explosives 8vo,  4  00 

Goodyear's  Coal-mines  of  the  Western   Coast  of  the   United 

States 12mo,  2  50 

Ihlseng's  Manual  of  Mining 8vo,  4  00 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe. 8vo,  1  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  00 

Sawyer's  Accidents  in  Mines 8vo,  7  00 

Walke's  Lectures  on  Explosives 8vo,  4  00 

Wilson's  Cyanide  Processes 12mo,  1  50 

Wilson's  Chlorination  Process 12mo,  1  50 

Wilaon's  Hydraulic  and  Placer  Mining 12mo,  2  00 

Wilsen's  Treatise  on  Practical  and  Theoretical  Mine  Ventila- 
tion   12mo,  1  25 

15 


SANITARY  SCIENCE. 

Fol well's  Sewerage.    (Designing,  Construction  and  Maintenance.) 

8vo,  3  00 

Water-supply    Engineering 8vo,  4  00 

Fuertes's  Water  and  Public  Health 12mo,  1  50 

Water-filtration   Works 12mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection 16mo,  1  00 

Goodrich's  Economical  Disposal  of  Towns'  Refuse ...  Demy  8vo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  00 

Kiersted's  Sewage  Disposal , 12mo,  1  25 

Mason's  Water-supply.     (Considered   Principally  from  a  San- 
itary Standpoint 8vo,  5  00 

"        Examination    of    Water.      (Chemical    and    Bacterio- 
logical.)     12mo,  1  25 

Merriman's  Elements  of  Sanitary  Engineering 8vo,  2  00 

Nichols's  Water-supply.     (Considered  Mainly  from  a  Chemical 

and  Sanitary  Standpoint.)     (1883.)  8vo,  2  50 

Ogden's  Sewer  Design 12mo,  2  00 

*  Price's  Handbook  on  Sanitation 12mo,  1  50 

Richards's  Cost  of  Food.    A  Study  in  Dietaries 12mo,  1  00 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sani- 
tary   Standpoint 8vo,  2  00 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science .  12mo,  1  00 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage! 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

WoodhulPs  Notes  on  Military  Hygiene 16mo,  1  50 

MISCELLANEOUS. 

Barker's  Deep-sea  Soundings 8vo,  2  00 

Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Ex- 
cursion   of    the    International    Congress    of    Geologists. 

Large  8vo,  1  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo,  4  00 

Haines's  American  Railway  Management 12mo,  2  50 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food. 

Mounted  chart,  1  25 

"      Fallacy  of  the  Present  Theory  of  Sound 16mo,  1  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824- 

1894 Small    8vo,  3  00 

Rotherham's  Emphasised  New  Testament Large  8vo,  2  00 

"            Critical  Emphasised  New  Testament 12mo,  1  50 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Totten's  Important  Question  in  Metrology 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  1  00 

Worcester  and  Atkinson.    Small  Hospitals,  Establishment  and 
Maintenance,  and  Suggestions  for  Hospital  Architecture, 

with  Plans  for  a  Small  Hospital 12mo,  1  25 

HEBREW    AND    CHALDEE    TEXT-BOOKS. 

Green's  Grammar  of  the  Hebrew  Language 8vo,  3  00 

"       Elementary  Hebrew  Grammar 12mo,  1  25 

"       Hebrew  Chrestomathy 8vo,  2  00 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament 

Scriptures.     (Tregelles.) Small  4to,  half  morocco,  5  00 

Letteris's  Hebrew  Bible 8vo,  2  25 

16 


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YC   12714 


